%------------------------------------------------------------------------------
% File     : ITP289_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_DelImperative 00402_025871
% 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    : 0095_VEBT_DelImperative_00402_025871 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12579 (3823 unt;2007 typ;   0 def)
%            Number of atoms       : 26687 (9418 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 17889 (2793   ~; 581   |;1322   &)
%                                         (2903 <=>;10290  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   26 (   2 avg)
%            Number of FOOLs       : 1934 ( 713 fml;1221 var)
%            Number of X terms     : 1065 (   0  []; 797 ite; 268 let)
%            Number of types       :  419 ( 418 usr)
%            Number of type conns  : 1906 (1180   >; 726   *;   0   +;   0  <<)
%            Number of predicates  :  120 ( 117 usr;   2 prp; 0-3 aty)
%            Number of functors    : 1493 (1493 usr; 430 con; 0-4 aty)
%            Number of variables   : 26955 (26034   !; 921   ?;26955   :)
% SPC      : TX0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 22:12:25.357
%------------------------------------------------------------------------------
% Could-be-implicit typings (418)
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tff(sy_c_ATP_058Lamm__ax____,type,
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tff(sy_c_ATP_058Lamm__ay____,type,
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tff(sy_c_ATP_058Lamm__bb____,type,
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tff(sy_c_ATP_058Lamm__bc____,type,
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tff(sy_c_ATP_058Lamm__bd____,type,
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tff(sy_c_ATP_058Lamm__bf____,type,
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    aTP_Lamm_bw: complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: product_prod_int_int > fun_se4767462807232413326_int_o ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: vEBT_VEBT > fun_fu6329627000648617163VEBT_o ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: real > fun_fu9050450510664235267real_o ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: $o > fun_fun_o_o_fun_o_o ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: int > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: complex > fun_fu1358370060594899587plex_o ).

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

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_complex_o ).

tff(sy_c_ATP_058Lamm__cg____,type,
    aTP_Lamm_cg: fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_real_o ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_o_o ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_nat_o ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_int_o ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_real_real ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_int_int ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: fun_assn_assn ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__cr____,type,
    aTP_Lamm_cr: fun_real_real ).

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: fun_int_int ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cx____,type,
    aTP_Lamm_cx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: assn > fun_assn_assn ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_o_assn > fun_assn_fun_o_assn ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: fun_VEBT_VEBTi_assn > fun_as315122898794150911i_assn ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: fun_option_nat_assn > fun_as2043416024874063466t_assn ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_nat_assn > fun_as8240436605320707994t_assn ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_VE1571388752190420990T_assn > fun_li281471924019390937o_assn ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: fun_VE4931415397359990460t_assn > fun_li1885941422112166061o_assn ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_re8050108611700907676T_assn > fun_li4956905047377384685o_assn ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_re560840401387337987i_assn > fun_li5109120709684150270o_assn ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_re574865750389657758t_assn > fun_li2911986356186861721o_assn ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_o_4353168661238401050T_assn > fun_li4690688603795389557o_assn ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_o_9029690514724572677i_assn > fun_li8507508749202587254o_assn ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_o_fun_nat_assn > fun_li3743109713932019473o_assn ).

tff(sy_c_ATP_058Lamm__dm____,type,
    aTP_Lamm_dm: fun_na4489259242115493880T_assn > fun_li7316139697933403593o_assn ).

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_na6467670216208870567i_assn > fun_li7374752162181608482o_assn ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: vEBT_VEBT > fun_VE9201042017749490615o_assn ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: nat > fun_li3871283220525210964t_assn ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: vEBT_VEBT > fun_VE6841497200278198903t_assn ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_VE597728521808591204at_nat ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_real_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: fun_o_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_VE4436863797057549263at_nat ).

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

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: nat > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_VEBT_VEBT_nat > fun_VEBT_VEBT_real ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: nat > fun_li4611866707255273568o_assn ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: nat > fun_li6689927339917241139i_assn ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: nat > fun_li7646410839599160030t_assn ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: nat > fun_li6911922195568332558t_assn ).

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

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: assn > fun_VE2561185598974572172i_assn ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: assn > fun_op6483474040057813218t_assn ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: assn > fun_nat_fun_nat_assn ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: fun_o_assn > fun_assn_fun_o_assn ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: fun_VEBT_VEBTi_assn > fun_as315122898794150911i_assn ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_option_nat_assn > fun_as2043416024874063466t_assn ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_nat_assn > fun_as8240436605320707994t_assn ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: list_VEBT_VEBT > fun_ar1453094831611173103i_assn ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: assn > fun_li3352006551604695530i_assn ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: assn > fun_fu5741754899130534993i_assn ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_li3352006551604695530i_assn > fun_as5323700489645809679i_assn ).

tff(sy_c_ATP_058Lamm__eu____,type,
    aTP_Lamm_eu: set_int > fun_se2051630309975736070_int_o ).

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: set_Code_integer > fun_se5529282185252607396eger_o ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: set_complex > fun_se823489619503225858plex_o ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: set_nat > fun_se2170385732985038782_nat_o ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_nat_o > fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: set_Code_integer > fun_na1863607095336112845eger_o ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: set_real > fun_na5739727354687878078real_o ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: set_VEBT_VEBTi > fun_na7469306952674514293EBTi_o ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: set_int > fun_na7306406234920378942_int_o ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: set_Code_integer > fun_na1863607095336112845eger_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: set_complex > fun_na2418085665696872764plex_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: set_VEBT_VEBT > fun_na5873943405026509934VEBT_o ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: set_real > fun_na5739727354687878078real_o ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: set_o > fun_nat_fun_list_o_o ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: set_VEBT_VEBTi > fun_na7469306952674514293EBTi_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: set_nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_nat_fun_real_o ).

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

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: set_Code_integer > fun_se7289222521248229150eger_o ).

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

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: set_nat > fun_set_nat_o ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: set_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__gd____,type,
    aTP_Lamm_gd: set_VEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: set_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: set_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: set_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: set_Pr958786334691620121nt_int > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: fun_nat_o > fun_nat_o ).

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

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

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: fun_Pr3238374608701176162_int_o > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: fun_Pr5160692531401422620nt_nat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: fun_Pr5160692531401422620nt_nat > fun_fu7601549123499533073_int_o ).

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

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_nat_nat > fun_fu5477517762487726877_nat_o ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: fun_int_nat > fun_int_o ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_int_nat > fun_fu1805223966450871909_int_o ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: fun_Code_integer_nat > fun_Code_integer_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_Code_integer_nat > fun_fu2182969643484777475eger_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: fun_complex_nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_complex_nat > fun_fu6122667721776870753plex_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: fun_Pr5160692531401422620nt_nat > fun_Pr9018117161326028078_int_o ).

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

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: fun_int_nat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_Code_integer_nat > fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_complex_nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: fun_re4238438383274954274t_real ).

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

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

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_Pr5160692531401422620nt_nat > fun_fu3068557634114553409_int_o ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_nat_nat > fun_fu4001317426178549763_nat_o ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_int_nat > fun_fu3883539188510166275_int_o ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: fun_Code_integer_nat > fun_fu8094397175230428611eger_o ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: fun_complex_nat > fun_fu1358370060594899587plex_o ).

tff(sy_c_ATP_058Lamm__hj____,type,
    aTP_Lamm_hj: fun_nat_complex ).

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: fun_nat_real ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: fun_nat_int ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: fun_nat_real ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: nat > fun_fu40286998686754419omplex ).

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

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__hs____,type,
    aTP_Lamm_hs: nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: complex > fun_fu40286998686754419omplex ).

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

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_nat_o > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_nat_o > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_nat_o > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_nat_o > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: set_nat > fun_fu40286998686754419omplex ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: set_nat > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: set_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: set_nat > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ij____,type,
    aTP_Lamm_ij: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_nat_real > fun_re4238438383274954274t_real ).

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

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

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__iz____,type,
    aTP_Lamm_iz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jc____,type,
    aTP_Lamm_jc: fun_nat_real ).

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

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jo____,type,
    aTP_Lamm_jo: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__jp____,type,
    aTP_Lamm_jp: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__jq____,type,
    aTP_Lamm_jq: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: num > fun_na4138050610385705875at_nat ).

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

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_na4138050610385705875at_nat ).

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

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__kd____,type,
    aTP_Lamm_kd: fun_real_o ).

tff(sy_c_ATP_058Lamm__ke____,type,
    aTP_Lamm_ke: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__kf____,type,
    aTP_Lamm_kf: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__kg____,type,
    aTP_Lamm_kg: fun_int_o ).

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ki____,type,
    aTP_Lamm_ki: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: int > fun_int_int ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: code_integer > fun_Co8499238673658658103nteger ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__km____,type,
    aTP_Lamm_km: real > fun_int_o ).

tff(sy_c_ATP_058Lamm__kn____,type,
    aTP_Lamm_kn: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__ko____,type,
    aTP_Lamm_ko: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__kp____,type,
    aTP_Lamm_kp: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__kq____,type,
    aTP_Lamm_kq: int > fun_in5570076352459579423_int_o ).

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

tff(sy_c_ATP_058Lamm__ks____,type,
    aTP_Lamm_ks: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__kt____,type,
    aTP_Lamm_kt: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__ku____,type,
    aTP_Lamm_ku: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__kv____,type,
    aTP_Lamm_kv: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__kw____,type,
    aTP_Lamm_kw: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__kx____,type,
    aTP_Lamm_kx: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ky____,type,
    aTP_Lamm_ky: fun_real_real > fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__kz____,type,
    aTP_Lamm_kz: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__la____,type,
    aTP_Lamm_la: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lb____,type,
    aTP_Lamm_lb: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__lc____,type,
    aTP_Lamm_lc: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__ld____,type,
    aTP_Lamm_ld: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__le____,type,
    aTP_Lamm_le: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__lf____,type,
    aTP_Lamm_lf: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__lg____,type,
    aTP_Lamm_lg: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__lh____,type,
    aTP_Lamm_lh: fun_real_real ).

tff(sy_c_ATP_058Lamm__li____,type,
    aTP_Lamm_li: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lj____,type,
    aTP_Lamm_lj: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lk____,type,
    aTP_Lamm_lk: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ll____,type,
    aTP_Lamm_ll: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__lm____,type,
    aTP_Lamm_lm: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ln____,type,
    aTP_Lamm_ln: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__lo____,type,
    aTP_Lamm_lo: fun_real_real ).

tff(sy_c_ATP_058Lamm__lp____,type,
    aTP_Lamm_lp: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lq____,type,
    aTP_Lamm_lq: fun_nat_real ).

tff(sy_c_ATP_058Lamm__lr____,type,
    aTP_Lamm_lr: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ls____,type,
    aTP_Lamm_ls: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__lt____,type,
    aTP_Lamm_lt: fun_nat_real ).

tff(sy_c_ATP_058Lamm__lu____,type,
    aTP_Lamm_lu: fun_nat_real ).

tff(sy_c_ATP_058Lamm__lv____,type,
    aTP_Lamm_lv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lw____,type,
    aTP_Lamm_lw: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lx____,type,
    aTP_Lamm_lx: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ly____,type,
    aTP_Lamm_ly: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__lz____,type,
    aTP_Lamm_lz: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ma____,type,
    aTP_Lamm_ma: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mb____,type,
    aTP_Lamm_mb: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__mc____,type,
    aTP_Lamm_mc: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__md____,type,
    aTP_Lamm_md: fun_nat_real > fun_fu6119738925743952495t_real ).

tff(sy_c_ATP_058Lamm__me____,type,
    aTP_Lamm_me: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mf____,type,
    aTP_Lamm_mf: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mg____,type,
    aTP_Lamm_mg: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__mh____,type,
    aTP_Lamm_mh: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__mi____,type,
    aTP_Lamm_mi: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__mj____,type,
    aTP_Lamm_mj: fun_real_real ).

tff(sy_c_ATP_058Lamm__mk____,type,
    aTP_Lamm_mk: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ml____,type,
    aTP_Lamm_ml: fun_re7681423839302404990l_real ).

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

tff(sy_c_ATP_058Lamm__mn____,type,
    aTP_Lamm_mn: fun_real_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__mo____,type,
    aTP_Lamm_mo: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__mp____,type,
    aTP_Lamm_mp: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__mq____,type,
    aTP_Lamm_mq: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__mr____,type,
    aTP_Lamm_mr: fun_real_o > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ms____,type,
    aTP_Lamm_ms: real > fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__mt____,type,
    aTP_Lamm_mt: fun_real_o > fun_real_o ).

tff(sy_c_ATP_058Lamm__mu____,type,
    aTP_Lamm_mu: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__mv____,type,
    aTP_Lamm_mv: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__mw____,type,
    aTP_Lamm_mw: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__mx____,type,
    aTP_Lamm_mx: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__my____,type,
    aTP_Lamm_my: fun_Co3583066429914935172er_nat ).

tff(sy_c_ATP_058Lamm__mz____,type,
    aTP_Lamm_mz: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__na____,type,
    aTP_Lamm_na: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: fun_real_int ).

tff(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
    archim2898591450579166408c_real: fun_real_real ).

tff(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: fun_rat_int ).

tff(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: fun_real_int ).

tff(sy_c_Array__Time_Onth_001_Eo,type,
    array_nth_o: ( array_o * nat ) > heap_Time_Heap_o ).

tff(sy_c_Array__Time_Onth_001t__Nat__Onat,type,
    array_nth_nat: ( array_nat * nat ) > heap_Time_Heap_nat ).

tff(sy_c_Array__Time_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
    array_nth_option_nat: ( array_option_nat * nat ) > heap_T2636463487746394924on_nat ).

tff(sy_c_Array__Time_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_nth_VEBT_VEBTi: ( array_VEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > fun_Pr6460944699146269935_nat_o ).

tff(sy_c_Assertions_Oentails,type,
    entails: ( assn * assn ) > $o ).

tff(sy_c_Assertions_Oex__assn_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ex_ass463751140784270563_VEBTi: fun_li3352006551604695530i_assn > assn ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: fun_o_assn ).

tff(sy_c_Assertions_Osnga__assn_001_Eo,type,
    snga_assn_o: ( array_o * list_o ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__Nat__Onat,type,
    snga_assn_nat: ( array_nat * list_nat ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__Option__Ooption_It__Nat__Onat_J,type,
    snga_assn_option_nat: ( array_option_nat * list_option_nat ) > assn ).

tff(sy_c_Assertions_Osnga__assn_001t__VEBT____BuildupMemImp__OVEBTi,type,
    snga_assn_VEBT_VEBTi: array_VEBT_VEBTi > fun_li3352006551604695530i_assn ).

tff(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: ( assn * assn * assn ) > $o ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Bit__Operations_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 > fun_num_num ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
    bit_ri7632146776885996613nteger: code_integer > code_integer ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: fun_int_int ).

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

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > fun_int_int ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
    bit_se3928097537394005634nteger: ( nat * code_integer ) > code_integer ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: nat > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: fun_nat_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: fun_nat_nat ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: int > fun_int_int ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: fun_nat_fun_nat_nat ).

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tff(sy_c_List_Oupto__aux,type,
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tff(sy_c_List_Oupto__rel,type,
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tff(sy_c_Nat_OSuc,type,
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tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
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tff(sy_c_Nat__Bijection_Oset__encode,type,
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tff(sy_c_Nat__Bijection_Otriangle,type,
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tff(sy_c_NthRoot_Oroot,type,
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tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Real__Oreal_M_Eo_J,type,
    bot_bot_fun_real_o: fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
    ord_less_o: $o > fun_o_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: fun_Co2081214437782444048eger_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: fun_num_fun_num_o ).

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_less_option_nat: fun_op5794009454290973322_nat_o ).

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

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

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: fun_real_fun_real_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
    ord_less_set_o: set_o > fun_set_o_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > fun_set_complex_o ).

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

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

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

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

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Real__Oreal_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Rat__Orat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ord_le6592769550269828683_VEBTi: ( set_VEBT_VEBTi * set_VEBT_VEBTi ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Int__Oint_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Nat__Onat_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__Real__Oreal_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Oless__eq_001t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
    ord_max_int: int > fun_int_int ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
    ord_max_num: num > fun_num_num ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
    ord_max_real: fun_re7681423839302404990l_real ).

tff(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_I_Eo_J,type,
    ord_max_set_o: ( set_o * set_o ) > set_o ).

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

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

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

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

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__Numeral____Type__Onum0_J,type,
    top_to3689904424835650196l_num0: set_Numeral_num0 ).

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__Oliteral_J,type,
    top_top_set_literal: set_literal ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
    power_8256067586552552935nteger: ( code_integer * nat ) > code_integer ).

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

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

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

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

tff(sy_c_Product__Type_OPair_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc7507926704131184380et_nat: ( heap_e7401611519738050253t_unit * set_nat ) > produc3658429121746597890et_nat ).

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

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

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

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
    produc4511245868158468465l_real: ( real * real ) > produc2422161461964618553l_real ).

tff(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc537772716801021591T_VEBT: ( vEBT_VEBT * vEBT_VEBT ) > produc8243902056947475879T_VEBT ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
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    time_htt_o: ( assn * heap_Time_Heap_o * fun_o_assn ) > fun_nat_o ).

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

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

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

tff(sy_c_Time__Reasoning_Otime_001_Eo,type,
    time_time_o: ( heap_Time_Heap_o * heap_e7401611519738050253t_unit ) > nat ).

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

tff(sy_c_Time__Reasoning_Otime_001t__Nat__Onat,type,
    time_time_nat: ( heap_Time_Heap_nat * heap_e7401611519738050253t_unit ) > nat ).

tff(sy_c_Time__Reasoning_Otime_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_time_option_nat: ( heap_T2636463487746394924on_nat * heap_e7401611519738050253t_unit ) > nat ).

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

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

tff(sy_c_Topological__Spaces_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__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

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

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

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

tff(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum0,type,
    type_l4264026598287037464l_num0: itself_Numeral_num0 > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: fun_nat_int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: fun_nat_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun_nat_fun_nat_o ).

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

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

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

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli__rel,type,
    vEBT_V5740978063120863272li_rel: fun_VE8636387394336989792EBTi_o ).

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

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

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

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

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

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

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

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: $o > fun_o_VEBT_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: ( option4927543243414619207at_nat * nat * array_VEBT_VEBTi ) > fun_VE8449866691444886503_VEBTi ).

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

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    vEBT_c1335663792808957512ap_nat: ( fun_op2425996437410981688ap_nat * fun_o_7009304396571837746ap_nat * vEBT_VEBTi ) > heap_Time_Heap_nat ).

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

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

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Nat__Onat,type,
    vEBT_case_VEBTi_nat: ( fun_op7256438278201533260Ti_nat * fun_o_fun_o_nat ) > fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: fun_VEBT_VEBTi_nat ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun_VE905575822219544865i_assn ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: fun_Pr3720955412185915546EBTi_o ).

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

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

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

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: fun_VE8636387394336989792EBTi_o ).

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

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

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: fun_VE8636387394336989792EBTi_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option4927543243414619207at_nat * nat * list_VEBT_VEBT ) > fun_VE3165209603783583043T_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: fun_VE6480751511521678896_nat_o ).

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

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: fun_nat_VEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__DelImperative_OVEBT__internal_Ovebt__deletei_H,type,
    vEBT_V1365221501068881998eletei: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_T8145700208782473153_VEBTi ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: vEBT_VEBT > fun_nat_nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: vEBT_VEBT > fun_nat_VEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001_Eo_001t__Nat__Onat,type,
    vEBT_L2281750874075065672_o_nat: ( set_nat * fun_o_fun_nat_assn * list_o * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6286945158656146733_VEBTi: ( set_nat * fun_o_9029690514724572677i_assn * list_o ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1319876754960170684T_VEBT: ( set_nat * fun_o_4353168661238401050T_assn * list_o * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7489483478785760935_VEBTi: ( set_nat * fun_na6467670216208870567i_assn * list_nat ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8511957252848910786T_VEBT: ( set_nat * fun_na4489259242115493880T_assn * list_nat * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001_Eo,type,
    vEBT_L7980206306069228746real_o: ( set_nat * fun_real_fun_o_assn * list_real * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L234762979517870878al_nat: ( set_nat * fun_re574865750389657758t_assn * list_real * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__Real__Oreal,type,
    vEBT_L5184575500739366650l_real: ( set_nat * fun_re6303261851654292474l_assn * list_real * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7851252805511451907_VEBTi: ( set_nat * fun_re560840401387337987i_assn * list_real ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3095048238742455910T_VEBT: ( set_nat * fun_re8050108611700907676T_assn * list_real * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L886525131989349516_VEBTi: ( set_nat * fun_VE2561185598974572172i_assn * list_VEBT_VEBTi ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2497118539674116125T_VEBT: ( set_nat * fun_VE5577048379444288083T_assn * list_VEBT_VEBTi * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    vEBT_L7058566406413635588VEBT_o: ( set_nat * fun_VE2104784059010244834o_assn * list_VEBT_VEBT * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8650695023172932196BT_nat: ( set_nat * fun_VE4931415397359990460t_assn * list_VEBT_VEBT * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L4281036506115550016T_real: ( set_nat * fun_VE6910866152645720856l_assn * list_VEBT_VEBT * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: ( set_nat * fun_VE905575822219544865i_assn * list_VEBT_VEBT ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3204528365124325536T_VEBT: ( set_nat * fun_VE1571388752190420990T_assn * list_VEBT_VEBT * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_OlistI__assn_001tf__c____11__058ATP_001tf__d____11__058ATP,type,
    vEBT_L375988980963497884911_ATP: ( set_nat * fun_c_1625295384132212607P_assn * list_c_11_ATP * list_d_11_ATP ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Nat__Onat,type,
    vEBT_L4785011123346445925_o_nat: ( fun_o_fun_nat_assn * list_o * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Real__Oreal,type,
    vEBT_L4725278957065240257o_real: ( fun_o_fun_real_assn * list_o * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L3704169666673096010_VEBTi: ( fun_o_9029690514724572677i_assn * list_o ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1750719106661372127T_VEBT: ( fun_o_4353168661238401050T_assn * list_o * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    vEBT_L4260503343685368993omplex: ( fun_co6163879640491479806x_assn * list_complex * list_complex ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Int__Oint,type,
    vEBT_L134985006839036959ex_int: ( fun_co6164383388265235324t_assn * list_complex * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    vEBT_L137475477348087235ex_nat: ( fun_co1748589094918083360t_assn * list_complex * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    vEBT_L2479436891206192927x_real: ( fun_co8000059489268409724l_assn * list_complex * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8524933119956041985T_VEBT: ( fun_co2031555705788908698T_assn * list_complex * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Complex__Ocomplex,type,
    vEBT_L6716599654974302751omplex: ( fun_in9020965360991744636x_assn * list_int * list_complex ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Int__Oint,type,
    vEBT_L74593716426352029nt_int: ( fun_int_fun_int_assn * list_int * list_int ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Nat__Onat,type,
    vEBT_L77084186935402305nt_nat: ( fun_int_fun_nat_assn * list_int * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Real__Oreal,type,
    vEBT_L8288995350762215837t_real: ( fun_in6446958447386763770l_assn * list_int * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1664421287176695555T_VEBT: ( fun_in7755481285845205148T_assn * list_int * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Real__Oreal,type,
    vEBT_L6102073776069194049t_real: ( fun_na448437756743965086l_assn * list_nat * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L4387162340545308618_VEBTi: ( fun_na6467670216208870567i_assn * list_nat ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8158188754432654943T_VEBT: ( fun_na4489259242115493880T_assn * list_nat * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001_Eo,type,
    vEBT_L6234343332106409831real_o: ( fun_real_fun_o_assn * list_real * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L1446010312343316929al_nat: ( fun_re574865750389657758t_assn * list_real * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Real__Oreal,type,
    vEBT_L1930518968523514909l_real: ( fun_re6303261851654292474l_assn * list_real * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L9060850011106065574_VEBTi: ( fun_re560840401387337987i_assn * list_real ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L4595930785310033027T_VEBT: ( fun_re8050108611700907676T_assn * list_real * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    vEBT_L8937798142398754470i_real: ( fun_VE1885522831824361987l_assn * list_VEBT_VEBTi * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L7265847600308530106T_VEBT: ( fun_VE5577048379444288083T_assn * list_VEBT_VEBTi * list_VEBT_VEBT ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    vEBT_L7489408758114837031VEBT_o: ( fun_VE2104784059010244834o_assn * list_VEBT_VEBT * list_o ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8296926524756676353BT_nat: ( fun_VE4931415397359990460t_assn * list_VEBT_VEBT * list_nat ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L5781919052683127133T_real: ( fun_VE6910866152645720856l_assn * list_VEBT_VEBT * list_real ) > assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6296928887356842470_VEBTi: ( fun_VE905575822219544865i_assn * list_VEBT_VEBT ) > fun_li3352006551604695530i_assn ).

tff(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1279224858307276611T_VEBT: ( fun_VE1571388752190420990T_assn * list_VEBT_VEBT * list_VEBT_VEBT ) > assn ).

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

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: fun_VE6480751511521678896_nat_o ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: fun_op5794009454290973322_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > fun_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: set_nat > fun_nat_o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
    vEBT_V4262088993061758097ft_nat: fun_nat_fun_nat_nat > fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option_nat ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun_Pr8877776288780692068_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: fun_nat_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun_nat_fun_nat_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun_VEBT_VEBT_real ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun_VEBT_VEBT_nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun_VE2962686792283380846VEBT_o ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun_VE2962686792283380846VEBT_o ).

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

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

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

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

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

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
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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__Num__Onum,type,
    aa_VEBT_VEBT_num: ( fun_VEBT_VEBT_num * vEBT_VEBT ) > num ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__Rat__Orat,type,
    aa_VEBT_VEBT_rat: ( fun_VEBT_VEBT_rat * vEBT_VEBT ) > rat ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    aa_VEBT_VEBT_real: ( fun_VEBT_VEBT_real * vEBT_VEBT ) > real ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    aa_VEB3973023575446460973_VEBTi: ( fun_VE712148701811721852_VEBTi * vEBT_VEBT ) > vEBT_VEBTi ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    aa_VEB5832953974749077948T_VEBT: ( fun_VE3165209603783583043T_VEBT * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
    aa_VEB5185540838851036891o_assn: ( fun_VE2104784059010244834o_assn * vEBT_VEBT ) > fun_o_assn ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_VEB2314620471664397097_nat_o: ( fun_VE6480751511521678896_nat_o * vEBT_VEBT ) > fun_nat_o ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__fun_It__Nat__Onat_Mt__Assertions__Oassn_J,type,
    aa_VEB8640158415945897581t_assn: ( fun_VE4931415397359990460t_assn * vEBT_VEBT ) > fun_nat_assn ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J,type,
    aa_VEB1122092716774696213at_nat: ( fun_VE597728521808591204at_nat * vEBT_VEBT ) > fun_nat_nat ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__fun_It__Real__Oreal_Mt__Assertions__Oassn_J,type,
    aa_VEB8575627266564790985l_assn: ( fun_VE6910866152645720856l_assn * vEBT_VEBT ) > fun_real_assn ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__fun_It__VEBT____BuildupMemImp__OVEBTi_Mt__Assertions__Oassn_J,type,
    aa_VEB3717855479479287634i_assn: ( fun_VE905575822219544865i_assn * vEBT_VEBT ) > fun_VEBT_VEBTi_assn ).

tff(sy_c_aa_001t__VEBT____Definitions__OVEBT_001t__fun_It__VEBT____Definitions__OVEBT_Mt__Assertions__Oassn_J,type,
    aa_VEB115603943081001463T_assn: ( fun_VE1571388752190420990T_assn * vEBT_VEBT ) > fun_VEBT_VEBT_assn ).

tff(sy_c_aa_001t__fun_I_Eo_M_Eo_J_001t__fun_I_Eo_M_Eo_J,type,
    aa_fun_o_o_fun_o_o: ( fun_fun_o_o_fun_o_o * fun_o_o ) > fun_o_o ).

tff(sy_c_aa_001t__fun_I_Eo_Mt__Assertions__Oassn_J_001t__fun_I_Eo_Mt__Assertions__Oassn_J,type,
    aa_fun4536517337817438332o_assn: ( fun_fu704708528062463619o_assn * fun_o_assn ) > fun_o_assn ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_M_Eo_J_001t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
    aa_fun8214818745163655228eger_o: ( fun_fu8094397175230428611eger_o * fun_Code_integer_o ) > fun_Code_integer_o ).

tff(sy_c_aa_001t__fun_It__Code____Numeral__Ointeger_Mt__Nat__Onat_J_001t__fun_It__Code____Numeral__Ointeger_M_Eo_J,type,
    aa_fun4054834855839773292eger_o: ( fun_fu2182969643484777475eger_o * fun_Code_integer_nat ) > fun_Code_integer_o ).

tff(sy_c_aa_001t__fun_It__Complex__Ocomplex_M_Eo_J_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    aa_fun2514317545224092796plex_o: ( fun_fu1358370060594899587plex_o * fun_complex_o ) > fun_complex_o ).

tff(sy_c_aa_001t__fun_It__Complex__Ocomplex_Mt__Nat__Onat_J_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    aa_fun8723867720292580170plex_o: ( fun_fu6122667721776870753plex_o * fun_complex_nat ) > fun_complex_o ).

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    aa_fun5425778157628386638_int_o: ( fun_fu1805223966450871909_int_o * fun_int_nat ) > fun_int_o ).

tff(sy_c_aa_001t__fun_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__Assertions__Oassn_J_001t__fun_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_Mt__Assertions__Oassn_J,type,
    aa_fun6795544665460849810i_assn: ( fun_fu5741754899130534993i_assn * fun_li3352006551604695530i_assn ) > fun_li3352006551604695530i_assn ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_M_Eo_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
    aa_fun5937193402907943164_nat_o: ( fun_fu4001317426178549763_nat_o * fun_nat_o ) > fun_nat_o ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J_001t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
    aa_fun2955221450537884980omplex: ( fun_fu40286998686754419omplex * fun_nat_complex ) > fun_nat_complex ).

tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Int__Oint_J_001t__fun_It__Nat__Onat_Mt__Int__Oint_J,type,
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tff(sy_c_aa_001t__fun_It__Nat__Onat_Mt__Nat__Onat_J_001t__fun_It__Nat__Onat_M_Eo_J,type,
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    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__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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    aa_fun5349570537165229698_int_o: ( fun_fu3068557634114553409_int_o * fun_Pr3238374608701176162_int_o ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Nat__Onat_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    aa_fun6302685600795359042_int_o: ( fun_fu7601549123499533073_int_o * fun_Pr5160692531401422620nt_nat ) > fun_Pr3238374608701176162_int_o ).

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

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

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

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

tff(sy_c_aa_001tf__c____11__058ATP_001t__fun_Itf__d____11__058ATP_Mt__Assertions__Oassn_J,type,
    aa_c_13644421539718531832P_assn: ( fun_c_1625295384132212607P_assn * c_11_ATP ) > fun_d_11_ATP_assn ).

tff(sy_c_aa_001tf__d____11__058ATP_001t__Assertions__Oassn,type,
    aa_d_11_ATP_assn: ( fun_d_11_ATP_assn * d_11_ATP ) > assn ).

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

tff(sy_c_fequal_001_Eo,type,
    fequal_o: $o > fun_o_o ).

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

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_o ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_o ).

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

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_real_fun_real_o ).

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

tff(sy_c_member_001_Eo,type,
    member_o: $o > fun_set_o_o ).

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

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

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

tff(sy_c_member_001t__List__Olist_I_Eo_J,type,
    member_list_o: ( list_o * set_list_o ) > $o ).

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

tff(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
    member_list_real: ( list_real * set_list_real ) > $o ).

tff(sy_c_member_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    member8117914210271334748_VEBTi: ( list_VEBT_VEBTi * set_list_VEBT_VEBTi ) > $o ).

tff(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    member2936631157270082147T_VEBT: ( list_VEBT_VEBT * set_list_VEBT_VEBT ) > $o ).

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

tff(sy_c_member_001t__Num__Onum,type,
    member_num: num > fun_set_num_o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    member157494554546826820nteger: ( produc8923325533196201883nteger * set_Pr4811707699266497531nteger ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member6260224972018164377et_nat: ( produc3658429121746597890et_nat * set_Pr3948176798113811640et_nat ) > $o ).

tff(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: ( product_prod_int_int * set_Pr958786334691620121nt_int ) > $o ).

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

tff(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    member7279096912039735102um_num: ( product_prod_num_num * set_Pr8218934625190621173um_num ) > $o ).

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

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

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

tff(sy_c_member_001t__VEBT____BuildupMemImp__OVEBTi,type,
    member_VEBT_VEBTi: ( vEBT_VEBTi * set_VEBT_VEBTi ) > $o ).

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

tff(sy_v_A__11_058ATP,type,
    a_11_ATP: fun_c_1625295384132212607P_assn ).

tff(sy_v_F__11_058ATP,type,
    f_11_ATP: assn ).

tff(sy_v_I__11_058ATP,type,
    i_11_ATP: set_nat ).

tff(sy_v_aktnode____,type,
    aktnode: vEBT_VEBT ).

tff(sy_v_i__11_058ATP,type,
    i_11_ATP2: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_minew____,type,
    minew: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_newnode____,type,
    newnode: vEBT_VEBT ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_tree__is______,type,
    tree_is: list_VEBT_VEBTi ).

tff(sy_v_uu__16_058ATP,type,
    uu_16_ATP: list_c_11_ATP ).

tff(sy_v_uua__16_058ATP,type,
    uua_16_ATP: nat ).

tff(sy_v_va____,type,
    va: nat ).

tff(sy_v_x11______,type,
    x11: option4927543243414619207at_nat ).

tff(sy_v_x13______,type,
    x13: array_VEBT_VEBTi ).

tff(sy_v_x14______,type,
    x14: vEBT_VEBTi ).

tff(sy_v_xa____,type,
    xa: nat ).

tff(sy_v_xb______,type,
    xb: vEBT_VEBTi ).

tff(sy_v_xi__11_058ATP,type,
    xi_11_ATP: d_11_ATP ).

tff(sy_v_xnew____,type,
    xnew: nat ).

tff(sy_v_xs__11_058ATP,type,
    xs_11_ATP: list_c_11_ATP ).

tff(sy_v_xsi__11_058ATP,type,
    xsi_11_ATP: list_d_11_ATP ).

tff(sy_v_y______,type,
    y: nat ).

% Relevant facts (10546)
tff(fact_0_even__odd__cases,axiom,
    ! [Xc: nat] :
      ( ! [N: nat] : Xc != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),N)
     => ~ ! [N: nat] : Xc != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_nat_nat(suc,N)) ) ).

% even_odd_cases
tff(fact_1_groupy,axiom,
    ! [A: assn,B: assn,C: assn,D: assn,X: assn] :
      ( entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),aa_assn_assn(times_times_assn(C),D)),X)
     => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),C)),D),X) ) ).

% groupy
tff(fact_2_midextr,axiom,
    ! [P: assn,Q: assn,Q2: assn,R: assn,X: assn] :
      ( entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(P),Q)),Q2)),R),X)
     => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(P),R)),Q)),Q2),X) ) ).

% midextr
tff(fact_3_swappa,axiom,
    ! [B: assn,A: assn,C: assn,X: assn] :
      ( entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(B),A)),C),X)
     => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),C),X) ) ).

% swappa
tff(fact_4_power__shift,axiom,
    ! [Xc: nat,Ya: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),Ya) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(Xc)),some_nat(Ya)) = some_nat(Z) ) ) ).

% power_shift
tff(fact_5_bit__split__inv,axiom,
    ! [Xc: nat,D2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),D2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),D2)),D2) = Xc ).

% bit_split_inv
tff(fact_6_pow__sum,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ).

% pow_sum
tff(fact_7_mulcomm,axiom,
    ! [I: nat,Vaa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Vaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Vaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),I) ).

% mulcomm
tff(fact_8_high__def,axiom,
    ! [Xc: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% high_def
tff(fact_9_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ).

% high_bound_aux
tff(fact_10_high__inv,axiom,
    ! [Xc: nat,Nb: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xc)),Nb) = Ya ) ) ).

% high_inv
tff(fact_11_low__inv,axiom,
    ! [Xc: nat,Nb: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xc)),Nb) = Xc ) ) ).

% low_inv
tff(fact_12_minewdef,axiom,
    minew = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))) ).

% minewdef
tff(fact_13_bit__concat__def,axiom,
    ! [H: nat,L: nat,D2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(H),L),D2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),D2))),L) ).

% bit_concat_def
tff(fact_14_xndef,axiom,
    xnew = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))) ).

% xndef
tff(fact_15_newnodedef,axiom,
    newnode = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% newnodedef
tff(fact_16_local_Oext,axiom,
    ! [Ya: nat,TreeLista: list_VEBT_VEBT,X13a: array_VEBT_VEBTi,Tree_isa: list_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),size_s6755466524823107622T_VEBT(TreeLista))
     => entails(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_isa)),aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a)),aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),Ya)),nth_VEBT_VEBTi(Tree_isa,Ya))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(Ya,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_isa)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_isa)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(Ya,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_isa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),Ya)),nth_VEBT_VEBTi(Tree_isa,Ya)))) ) ).

% local.ext
tff(fact_17_aktnodedef,axiom,
    ( ( ma != mi )
   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,xa),ma)
     => ( aktnode = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% aktnodedef
tff(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062aktnode_O_A_I_092_060lbrakk_062ma_A_092_060noteq_062_Ami_059_Ax_A_092_060le_062_Ama_092_060rbrakk_062_A_092_060Longrightarrow_062_Aaktnode_A_061_AtreeList_A_B_Ahigh_A_I2_A_K_A2_A_094_A_Iva_Adiv_A2_J_A_K_Athe_A_Ivebt__mint_Asummary_J_A_L_Athe_A_Ivebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__mint_Asummary_J_J_J_J_A_ISuc_A_Iva_Adiv_A2_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Aktnode: vEBT_VEBT] :
        ~ ( ( ma != mi )
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,xa),ma)
           => ( Aktnode = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ).

% \<open>\<And>thesis. (\<And>aktnode. (\<lbrakk>ma \<noteq> mi; x \<le> ma\<rbrakk> \<Longrightarrow> aktnode = treeList ! high (2 * 2 ^ (va div 2) * the (vebt_mint summary) + the (vebt_mint (treeList ! the (vebt_mint summary)))) (Suc (va div 2))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_19_sum__power2__eq__zero__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( Xc = zero_zero_rat )
        & ( Ya = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_20_sum__power2__eq__zero__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( Xc = zero_zero_real )
        & ( Ya = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_21_sum__power2__eq__zero__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( ( aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( Xc = zero_zero_int )
        & ( Ya = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_22_sum__power2__eq__zero__iff,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( ( plus_p5714425477246183910nteger(power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger )
    <=> ( ( Xc = zero_z3403309356797280102nteger )
        & ( Ya = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_23_zero__less__power2,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A2 != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
tff(fact_24_zero__less__power2,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A2 != zero_zero_real ) ) ).

% zero_less_power2
tff(fact_25_zero__less__power2,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A2 != zero_zero_rat ) ) ).

% zero_less_power2
tff(fact_26_zero__less__power2,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( A2 != zero_zero_int ) ) ).

% zero_less_power2
tff(fact_27_add__self__div__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = M ).

% add_self_div_2
tff(fact_28_div2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_29_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc
tff(fact_30_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_31_zero__eq__power2,axiom,
    ! [A2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% zero_eq_power2
tff(fact_32_zero__eq__power2,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat )
    <=> ( A2 = zero_zero_nat ) ) ).

% zero_eq_power2
tff(fact_33_zero__eq__power2,axiom,
    ! [A2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% zero_eq_power2
tff(fact_34_zero__eq__power2,axiom,
    ! [A2: int] :
      ( ( aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% zero_eq_power2
tff(fact_35_zero__eq__power2,axiom,
    ! [A2: complex] :
      ( ( aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex )
    <=> ( A2 = zero_zero_complex ) ) ).

% zero_eq_power2
tff(fact_36_zero__eq__power2,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
tff(fact_37_tcd,axiom,
    ! [I: nat,TreeLista: list_VEBT_VEBT,TreeList: list_VEBT_VEBT,Ya: vEBT_VEBT,Xc: vEBT_VEBTi,X13a: array_VEBT_VEBTi,Tree_isa: list_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeLista))
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = size_s6755466524823107622T_VEBT(TreeList) )
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ya),Xc)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,I,Ya)),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,I,Ya)),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc)))) ) ) ).

% tcd
tff(fact_38_tcd,axiom,
    ! [I: nat,TreeLista: list_VEBT_VEBT,TreeList: list_real,Ya: vEBT_VEBT,Xc: vEBT_VEBTi,X13a: array_VEBT_VEBTi,Tree_isa: list_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeLista))
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = size_size_list_real(TreeList) )
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ya),Xc)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,I,Ya)),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,I,Ya)),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc)))) ) ) ).

% tcd
tff(fact_39_tcd,axiom,
    ! [I: nat,TreeLista: list_VEBT_VEBT,TreeList: list_o,Ya: vEBT_VEBT,Xc: vEBT_VEBTi,X13a: array_VEBT_VEBTi,Tree_isa: list_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeLista))
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = size_size_list_o(TreeList) )
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ya),Xc)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,I,Ya)),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,I,Ya)),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc)))) ) ) ).

% tcd
tff(fact_40_tcd,axiom,
    ! [I: nat,TreeLista: list_VEBT_VEBT,TreeList: list_nat,Ya: vEBT_VEBT,Xc: vEBT_VEBTi,X13a: array_VEBT_VEBTi,Tree_isa: list_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeLista))
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = size_size_list_nat(TreeList) )
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ya),Xc)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,I,Ya)),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,I,Ya)),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc)))) ) ) ).

% tcd
tff(fact_41_tcd,axiom,
    ! [I: nat,TreeLista: list_VEBT_VEBT,TreeList: list_VEBT_VEBTi,Ya: vEBT_VEBT,Xc: vEBT_VEBTi,X13a: array_VEBT_VEBTi,Tree_isa: list_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeLista))
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = size_s7982070591426661849_VEBTi(TreeList) )
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ya),Xc)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,I,Ya)),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,I,Ya)),list_u6098035379799741383_VEBTi(Tree_isa,I,Xc)))) ) ) ).

% tcd
tff(fact_42_recomp,axiom,
    ! [I: nat,TreeLista: list_VEBT_VEBT,Tree_isa: list_VEBT_VEBTi,X13a: array_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeLista))
     => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),I)),nth_VEBT_VEBTi(Tree_isa,I))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_isa))),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_isa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_isa))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_isa))) ) ).

% recomp
tff(fact_43_repack,axiom,
    ! [I: nat,TreeLista: list_VEBT_VEBT,Tree_isa: list_VEBT_VEBTi,Rest: assn,X13a: array_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(TreeLista))
     => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),I)),nth_VEBT_VEBTi(Tree_isa,I))),Rest)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_isa)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_isa))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(Rest),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_isa))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_isa))) ) ).

% repack
tff(fact_44_max__in__set__def,axiom,
    ! [Xs: set_nat,Xc: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),Xc)
    <=> ( aa_set_nat_o(member_nat(Xc),Xs)
        & ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Xc) ) ) ) ).

% max_in_set_def
tff(fact_45_min__in__set__def,axiom,
    ! [Xs: set_nat,Xc: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),Xc)
    <=> ( aa_set_nat_o(member_nat(Xc),Xs)
        & ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),X2) ) ) ) ).

% min_in_set_def
tff(fact_46_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( numera6690914467698888265omplex(M) = numera6690914467698888265omplex(Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_47_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,M) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_48_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,M) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_49_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,M) = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_50_numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,M) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( M = Nb ) ) ).

% numeral_eq_iff
tff(fact_51_assnle,axiom,
    ! [TreeLista: list_VEBT_VEBT,Tree_isa: list_VEBT_VEBTi,X13a: array_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] : entails(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_isa)),aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_isa)),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_isa))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_isa))) ).

% assnle
tff(fact_52_numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Nb) ) ).

% numeral_le_iff
tff(fact_53_numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Nb) ) ).

% numeral_le_iff
tff(fact_54_numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Nb) ) ).

% numeral_le_iff
tff(fact_55_numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Nb) ) ).

% numeral_le_iff
tff(fact_56_numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb) ) ).

% numeral_less_iff
tff(fact_57_numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb) ) ).

% numeral_less_iff
tff(fact_58_numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb) ) ).

% numeral_less_iff
tff(fact_59_numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb) ) ).

% numeral_less_iff
tff(fact_60_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_61_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_62_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_63_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_64_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W))),Z) ).

% add_numeral_left
tff(fact_65_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),Nb)) ).

% numeral_plus_numeral
tff(fact_66_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),Nb)) ).

% numeral_plus_numeral
tff(fact_67_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),Nb)) ).

% numeral_plus_numeral
tff(fact_68_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb)) ).

% numeral_plus_numeral
tff(fact_69_numeral__plus__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),Nb)) ).

% numeral_plus_numeral
tff(fact_70_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Z)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_71_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_72_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Z)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_73_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_74_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),Z)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_75_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(times_times_num(M),Nb)) ).

% numeral_times_numeral
tff(fact_76_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),Nb)) ).

% numeral_times_numeral
tff(fact_77_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),Nb)) ).

% numeral_times_numeral
tff(fact_78_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Nb)) ).

% numeral_times_numeral
tff(fact_79_numeral__times__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),Nb)) ).

% numeral_times_numeral
tff(fact_80_sum__squares__eq__zero__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya)) = zero_zero_real )
    <=> ( ( Xc = zero_zero_real )
        & ( Ya = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_81_sum__squares__eq__zero__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xc),Xc)),aa_rat_rat(times_times_rat(Ya),Ya)) = zero_zero_rat )
    <=> ( ( Xc = zero_zero_rat )
        & ( Ya = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_82_sum__squares__eq__zero__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xc),Xc)),aa_int_int(times_times_int(Ya),Ya)) = zero_zero_int )
    <=> ( ( Xc = zero_zero_int )
        & ( Ya = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_83_distrib__right__numeral,axiom,
    ! [A2: complex,B2: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A2),B2)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A2),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B2),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_84_distrib__right__numeral,axiom,
    ! [A2: real,B2: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_85_distrib__right__numeral,axiom,
    ! [A2: rat,B2: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A2),B2)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B2),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_86_distrib__right__numeral,axiom,
    ! [A2: nat,B2: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_num_nat(numeral_numeral_nat,V))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_87_distrib__right__numeral,axiom,
    ! [A2: int,B2: int,V: num] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A2),B2)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(B2),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_88_distrib__left__numeral,axiom,
    ! [V: num,B2: complex,C2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B2),C2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B2)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C2)) ).

% distrib_left_numeral
tff(fact_89_distrib__left__numeral,axiom,
    ! [V: num,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C2)) ).

% distrib_left_numeral
tff(fact_90_distrib__left__numeral,axiom,
    ! [V: num,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(plus_plus_rat(B2),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B2)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C2)) ).

% distrib_left_numeral
tff(fact_91_distrib__left__numeral,axiom,
    ! [V: num,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C2)) ).

% distrib_left_numeral
tff(fact_92_distrib__left__numeral,axiom,
    ! [V: num,B2: int,C2: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(plus_plus_int(B2),C2)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),B2)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),C2)) ).

% distrib_left_numeral
tff(fact_93_mem__Collect__eq,axiom,
    ! [A2: real,P: fun_real_o] :
      ( aa_set_real_o(member_real(A2),collect_real(P))
    <=> aa_real_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_94_mem__Collect__eq,axiom,
    ! [A2: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),collect_VEBT_VEBT(P))
    <=> aa_VEBT_VEBT_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_95_mem__Collect__eq,axiom,
    ! [A2: nat,P: fun_nat_o] :
      ( aa_set_nat_o(member_nat(A2),collect_nat(P))
    <=> aa_nat_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_96_mem__Collect__eq,axiom,
    ! [A2: int,P: fun_int_o] :
      ( aa_set_int_o(member_int(A2),collect_int(P))
    <=> aa_int_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_97_mem__Collect__eq,axiom,
    ! [A2: complex,P: fun_complex_o] :
      ( aa_set_complex_o(member_complex(A2),collect_complex(P))
    <=> aa_complex_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_98_mem__Collect__eq,axiom,
    ! [A2: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      ( member5262025264175285858nt_int(A2,collec213857154873943460nt_int(P))
    <=> aa_Pro4592068669816069595_int_o(P,A2) ) ).

% mem_Collect_eq
tff(fact_99_Collect__mem__eq,axiom,
    ! [A: set_real] : collect_real(aTP_Lamm_a(A)) = A ).

% Collect_mem_eq
tff(fact_100_Collect__mem__eq,axiom,
    ! [A: set_VEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_aa(A)) = A ).

% Collect_mem_eq
tff(fact_101_Collect__mem__eq,axiom,
    ! [A: set_nat] : collect_nat(aTP_Lamm_ab(A)) = A ).

% Collect_mem_eq
tff(fact_102_Collect__mem__eq,axiom,
    ! [A: set_int] : collect_int(aTP_Lamm_ac(A)) = A ).

% Collect_mem_eq
tff(fact_103_Collect__mem__eq,axiom,
    ! [A: set_complex] : collect_complex(aTP_Lamm_ad(A)) = A ).

% Collect_mem_eq
tff(fact_104_Collect__mem__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int] : collec213857154873943460nt_int(aTP_Lamm_ae(A)) = A ).

% Collect_mem_eq
tff(fact_105_Collect__cong,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P,X3)
        <=> aa_nat_o(Q,X3) )
     => ( collect_nat(P) = collect_nat(Q) ) ) ).

% Collect_cong
tff(fact_106_Collect__cong,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P,X3)
        <=> aa_int_o(Q,X3) )
     => ( collect_int(P) = collect_int(Q) ) ) ).

% Collect_cong
tff(fact_107_Collect__cong,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X3: complex] :
          ( aa_complex_o(P,X3)
        <=> aa_complex_o(Q,X3) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_108_Collect__cong,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X3)
        <=> aa_Pro4592068669816069595_int_o(Q,X3) )
     => ( collec213857154873943460nt_int(P) = collec213857154873943460nt_int(Q) ) ) ).

% Collect_cong
tff(fact_109_div__mult__mult1__if,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) = $ite(C2 = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ).

% div_mult_mult1_if
tff(fact_110_div__mult__mult1__if,axiom,
    ! [C2: int,A2: int,B2: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2)) = $ite(C2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ).

% div_mult_mult1_if
tff(fact_111_div__mult__mult2,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ).

% div_mult_mult2
tff(fact_112_div__mult__mult2,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ).

% div_mult_mult2
tff(fact_113_div__mult__mult1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ).

% div_mult_mult1
tff(fact_114_div__mult__mult1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ).

% div_mult_mult1
tff(fact_115_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: complex,C2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(minus_minus_complex(B2),C2)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B2)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C2)) ).

% right_diff_distrib_numeral
tff(fact_116_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C2)) ).

% right_diff_distrib_numeral
tff(fact_117_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),B2)),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),C2)) ).

% right_diff_distrib_numeral
tff(fact_118_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: int,C2: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(minus_minus_int(B2),C2)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),B2)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),C2)) ).

% right_diff_distrib_numeral
tff(fact_119_left__diff__distrib__numeral,axiom,
    ! [A2: complex,B2: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A2),B2)),numera6690914467698888265omplex(V)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A2),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B2),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_120_left__diff__distrib__numeral,axiom,
    ! [A2: real,B2: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),aa_num_real(numeral_numeral_real,V)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_121_left__diff__distrib__numeral,axiom,
    ! [A2: rat,B2: rat,V: num] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A2),B2)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(B2),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_122_left__diff__distrib__numeral,axiom,
    ! [A2: int,B2: int,V: num] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(A2),B2)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A2),aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(B2),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_123_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_nat_nat(suc,Nb)) = zero_zero_rat ).

% power_0_Suc
tff(fact_124_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) = zero_zero_nat ).

% power_0_Suc
tff(fact_125_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_nat_nat(suc,Nb)) = zero_zero_real ).

% power_0_Suc
tff(fact_126_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(zero_zero_int),aa_nat_nat(suc,Nb)) = zero_zero_int ).

% power_0_Suc
tff(fact_127_power__0__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_nat_nat(suc,Nb)) = zero_zero_complex ).

% power_0_Suc
tff(fact_128_power__0__Suc,axiom,
    ! [Nb: nat] : power_8256067586552552935nteger(zero_z3403309356797280102nteger,aa_nat_nat(suc,Nb)) = zero_z3403309356797280102nteger ).

% power_0_Suc
tff(fact_129_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_rat ).

% power_zero_numeral
tff(fact_130_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_nat ).

% power_zero_numeral
tff(fact_131_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_real ).

% power_zero_numeral
tff(fact_132_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_int ).

% power_zero_numeral
tff(fact_133_power__zero__numeral,axiom,
    ! [K: num] : aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,K)) = zero_zero_complex ).

% power_zero_numeral
tff(fact_134_power__zero__numeral,axiom,
    ! [K: num] : power_8256067586552552935nteger(zero_z3403309356797280102nteger,aa_num_nat(numeral_numeral_nat,K)) = zero_z3403309356797280102nteger ).

% power_zero_numeral
tff(fact_135_Suc__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Nb),one)) ).

% Suc_numeral
tff(fact_136_power__add__numeral2,axiom,
    ! [A2: complex,M: num,Nb: num,B2: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,M))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,Nb))),B2)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb)))),B2) ).

% power_add_numeral2
tff(fact_137_power__add__numeral2,axiom,
    ! [A2: code_integer,M: num,Nb: num,B2: code_integer] : times_3573771949741848930nteger(power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,M)),times_3573771949741848930nteger(power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,Nb)),B2)) = times_3573771949741848930nteger(power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb))),B2) ).

% power_add_numeral2
tff(fact_138_power__add__numeral2,axiom,
    ! [A2: real,M: num,Nb: num,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,M))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,Nb))),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb)))),B2) ).

% power_add_numeral2
tff(fact_139_power__add__numeral2,axiom,
    ! [A2: rat,M: num,Nb: num,B2: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,M))),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,Nb))),B2)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb)))),B2) ).

% power_add_numeral2
tff(fact_140_power__add__numeral2,axiom,
    ! [A2: nat,M: num,Nb: num,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,Nb))),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb)))),B2) ).

% power_add_numeral2
tff(fact_141_power__add__numeral2,axiom,
    ! [A2: int,M: num,Nb: num,B2: int] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,M))),aa_int_int(times_times_int(aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,Nb))),B2)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb)))),B2) ).

% power_add_numeral2
tff(fact_142_power__add__numeral2,axiom,
    ! [A2: assn,M: num,Nb: num,B2: assn] : aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),aa_num_nat(numeral_numeral_nat,M))),aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),aa_num_nat(numeral_numeral_nat,Nb))),B2)) = aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb)))),B2) ).

% power_add_numeral2
tff(fact_143_power__add__numeral,axiom,
    ! [A2: complex,M: num,Nb: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb))) ).

% power_add_numeral
tff(fact_144_power__add__numeral,axiom,
    ! [A2: code_integer,M: num,Nb: num] : times_3573771949741848930nteger(power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,M)),power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,Nb))) = power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb))) ).

% power_add_numeral
tff(fact_145_power__add__numeral,axiom,
    ! [A2: real,M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb))) ).

% power_add_numeral
tff(fact_146_power__add__numeral,axiom,
    ! [A2: rat,M: num,Nb: num] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb))) ).

% power_add_numeral
tff(fact_147_power__add__numeral,axiom,
    ! [A2: nat,M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb))) ).

% power_add_numeral
tff(fact_148_power__add__numeral,axiom,
    ! [A2: int,M: num,Nb: num] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb))) ).

% power_add_numeral
tff(fact_149_power__add__numeral,axiom,
    ! [A2: assn,M: num,Nb: num] : aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),aa_num_nat(numeral_numeral_nat,M))),aa_nat_assn(power_power_assn(A2),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_assn(power_power_assn(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(M),Nb))) ).

% power_add_numeral
tff(fact_150_power__Suc0__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_151_power__Suc0__right,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_152_power__Suc0__right,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_153_power__Suc0__right,axiom,
    ! [A2: complex] : aa_nat_complex(power_power_complex(A2),aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_154_power__Suc0__right,axiom,
    ! [A2: code_integer] : power_8256067586552552935nteger(A2,aa_nat_nat(suc,zero_zero_nat)) = A2 ).

% power_Suc0_right
tff(fact_155_div__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,zero_zero_nat)) = M ).

% div_by_Suc_0
tff(fact_156_div__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_157_nat__power__eq__Suc__0__iff,axiom,
    ! [Xc: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = zero_zero_nat )
        | ( Xc = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_158_power__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(suc,zero_zero_nat) ).

% power_Suc_0
tff(fact_159_nat__zero__less__power__iff,axiom,
    ! [Xc: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xc)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_160_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: complex,B2: complex,W: num] :
      ( ( A2 = aa_complex_complex(divide1717551699836669952omplex(B2),numera6690914467698888265omplex(W)) )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(times_times_complex(A2),numera6690914467698888265omplex(W)) = B2,A2 = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_161_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W)) = B2,A2 = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_162_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( A2 = aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W)) = B2,A2 = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_163_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: complex,W: num,A2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),numera6690914467698888265omplex(W)) = A2 )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(A2),numera6690914467698888265omplex(W)),A2 = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_164_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W)) = A2 )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W)),A2 = zero_zero_real) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_165_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W)) = A2 )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,B2 = aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W)),A2 = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_166_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W))),B2) ) ).

% le_divide_eq_numeral1(1)
tff(fact_167_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W))),B2) ) ).

% le_divide_eq_numeral1(1)
tff(fact_168_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W))),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_169_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W))),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_170_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W))),B2) ) ).

% less_divide_eq_numeral1(1)
tff(fact_171_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W))),B2) ) ).

% less_divide_eq_numeral1(1)
tff(fact_172_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W))),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_173_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W))),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_174_div__mult__self4,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self4
tff(fact_175_div__mult__self4,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),C2)),A2)),B2) = aa_int_int(plus_plus_int(C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self4
tff(fact_176_div__mult__self3,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self3
tff(fact_177_div__mult__self3,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(C2),B2)),A2)),B2) = aa_int_int(plus_plus_int(C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self3
tff(fact_178_div__mult__self2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self2
tff(fact_179_div__mult__self2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A2),aa_int_int(times_times_int(B2),C2))),B2) = aa_int_int(plus_plus_int(C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self2
tff(fact_180_div__mult__self1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_mult_self1
tff(fact_181_div__mult__self1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A2),aa_int_int(times_times_int(C2),B2))),B2) = aa_int_int(plus_plus_int(C2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_mult_self1
tff(fact_182_power__eq__0__iff,axiom,
    ! [A2: rat,Nb: nat] :
      ( ( aa_nat_rat(power_power_rat(A2),Nb) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_183_power__eq__0__iff,axiom,
    ! [A2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_184_power__eq__0__iff,axiom,
    ! [A2: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_185_power__eq__0__iff,axiom,
    ! [A2: int,Nb: nat] :
      ( ( aa_nat_int(power_power_int(A2),Nb) = zero_zero_int )
    <=> ( ( A2 = zero_zero_int )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_186_power__eq__0__iff,axiom,
    ! [A2: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(A2),Nb) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_complex )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_187_power__eq__0__iff,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( ( power_8256067586552552935nteger(A2,Nb) = zero_z3403309356797280102nteger )
    <=> ( ( A2 = zero_z3403309356797280102nteger )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% power_eq_0_iff
tff(fact_188_div__mult__self1__is__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),M)),Nb) = M ) ) ).

% div_mult_self1_is_m
tff(fact_189_div__mult__self__is__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)),Nb) = M ) ) ).

% div_mult_self_is_m
tff(fact_190_txe,axiom,
    ! [Ya: nat,TreeLista: list_VEBT_VEBT,Tree_isa: list_VEBT_VEBTi,X13a: array_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),size_s6755466524823107622T_VEBT(TreeLista))
     => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),Ya)),nth_VEBT_VEBTi(Tree_isa,Ya))),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_isa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(Ya,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_isa)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),Tree_isa))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_isa))) ) ).

% txe
tff(fact_191_lesseq__shift,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_lesseq,some_nat(Xc)),some_nat(Ya)) ) ).

% lesseq_shift
tff(fact_192_power__mono__iff,axiom,
    ! [A2: real,B2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_193_power__mono__iff,axiom,
    ! [A2: code_integer,B2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,Nb)),power_8256067586552552935nteger(B2,Nb))
          <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_194_power__mono__iff,axiom,
    ! [A2: rat,B2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),Nb)),aa_nat_rat(power_power_rat(B2),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_195_power__mono__iff,axiom,
    ! [A2: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_196_power__mono__iff,axiom,
    ! [A2: int,B2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(B2),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_197_power2__eq__iff__nonneg,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xc = Ya ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_198_power2__eq__iff__nonneg,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xc)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ya)
       => ( ( power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xc = Ya ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_199_power2__eq__iff__nonneg,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => ( ( aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xc = Ya ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_200_power2__eq__iff__nonneg,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xc)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ya)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xc = Ya ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_201_power2__eq__iff__nonneg,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => ( ( aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xc = Ya ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_202_power2__less__eq__zero__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real)
    <=> ( A2 = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
tff(fact_203_power2__less__eq__zero__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger)
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
tff(fact_204_power2__less__eq__zero__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat)
    <=> ( A2 = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
tff(fact_205_power2__less__eq__zero__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int)
    <=> ( A2 = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
tff(fact_206_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_207_add__One__commute,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),Nb) = aa_num_num(plus_plus_num(Nb),one) ).

% add_One_commute
tff(fact_208_le__numeral__extra_I3_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_zero_real) ).

% le_numeral_extra(3)
tff(fact_209_le__numeral__extra_I3_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_zero_rat) ).

% le_numeral_extra(3)
tff(fact_210_le__numeral__extra_I3_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_zero_nat) ).

% le_numeral_extra(3)
tff(fact_211_le__numeral__extra_I3_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),zero_zero_int) ).

% le_numeral_extra(3)
tff(fact_212_div__le__dividend,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),M) ).

% div_le_dividend
tff(fact_213_div__le__mono,axiom,
    ! [M: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K)) ) ).

% div_le_mono
tff(fact_214_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_le_zero
tff(fact_215_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_le_zero
tff(fact_216_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_le_zero
tff(fact_217_not__numeral__le__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_le_zero
tff(fact_218_zero__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_le_numeral
tff(fact_219_zero__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_le_numeral
tff(fact_220_zero__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_le_numeral
tff(fact_221_zero__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_le_numeral
tff(fact_222_zero__le__power,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ).

% zero_le_power
tff(fact_223_zero__le__power,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,Nb)) ) ).

% zero_le_power
tff(fact_224_zero__le__power,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),Nb)) ) ).

% zero_le_power
tff(fact_225_zero__le__power,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ) ).

% zero_le_power
tff(fact_226_zero__le__power,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),Nb)) ) ).

% zero_le_power
tff(fact_227_power__mono,axiom,
    ! [A2: real,B2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb)) ) ) ).

% power_mono
tff(fact_228_power__mono,axiom,
    ! [A2: code_integer,B2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,Nb)),power_8256067586552552935nteger(B2,Nb)) ) ) ).

% power_mono
tff(fact_229_power__mono,axiom,
    ! [A2: rat,B2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),Nb)),aa_nat_rat(power_power_rat(B2),Nb)) ) ) ).

% power_mono
tff(fact_230_power__mono,axiom,
    ! [A2: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb)) ) ) ).

% power_mono
tff(fact_231_power__mono,axiom,
    ! [A2: int,B2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(B2),Nb)) ) ) ).

% power_mono
tff(fact_232_Suc__div__le__mono,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,M)),Nb)) ).

% Suc_div_le_mono
tff(fact_233_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb))),M) ).

% times_div_less_eq_dividend
tff(fact_234_div__times__less__eq__dividend,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),Nb)),M) ).

% div_times_less_eq_dividend
tff(fact_235_sum__squares__le__zero__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya))),zero_zero_real)
    <=> ( ( Xc = zero_zero_real )
        & ( Ya = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_236_sum__squares__le__zero__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xc),Xc)),aa_rat_rat(times_times_rat(Ya),Ya))),zero_zero_rat)
    <=> ( ( Xc = zero_zero_rat )
        & ( Ya = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_237_sum__squares__le__zero__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xc),Xc)),aa_int_int(times_times_int(Ya),Ya))),zero_zero_int)
    <=> ( ( Xc = zero_zero_int )
        & ( Ya = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_238_power__less__imp__less__base,axiom,
    ! [A2: code_integer,Nb: nat,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(A2,Nb)),power_8256067586552552935nteger(B2,Nb))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2) ) ) ).

% power_less_imp_less_base
tff(fact_239_power__less__imp__less__base,axiom,
    ! [A2: real,Nb: nat,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% power_less_imp_less_base
tff(fact_240_power__less__imp__less__base,axiom,
    ! [A2: rat,Nb: nat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),Nb)),aa_nat_rat(power_power_rat(B2),Nb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% power_less_imp_less_base
tff(fact_241_power__less__imp__less__base,axiom,
    ! [A2: nat,Nb: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ) ).

% power_less_imp_less_base
tff(fact_242_power__less__imp__less__base,axiom,
    ! [A2: int,Nb: nat,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(B2),Nb))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ) ).

% power_less_imp_less_base
tff(fact_243_power__le__imp__le__base,axiom,
    ! [A2: real,Nb: nat,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_nat_nat(suc,Nb)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% power_le_imp_le_base
tff(fact_244_power__le__imp__le__base,axiom,
    ! [A2: code_integer,Nb: nat,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,aa_nat_nat(suc,Nb))),power_8256067586552552935nteger(B2,aa_nat_nat(suc,Nb)))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2) ) ) ).

% power_le_imp_le_base
tff(fact_245_power__le__imp__le__base,axiom,
    ! [A2: rat,Nb: nat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,Nb))),aa_nat_rat(power_power_rat(B2),aa_nat_nat(suc,Nb)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% power_le_imp_le_base
tff(fact_246_power__le__imp__le__base,axiom,
    ! [A2: nat,Nb: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(suc,Nb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ) ).

% power_le_imp_le_base
tff(fact_247_power__le__imp__le__base,axiom,
    ! [A2: int,Nb: nat,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,Nb))),aa_nat_int(power_power_int(B2),aa_nat_nat(suc,Nb)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ).

% power_le_imp_le_base
tff(fact_248_power__inject__base,axiom,
    ! [A2: real,Nb: nat,B2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_nat_nat(suc,Nb)) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_249_power__inject__base,axiom,
    ! [A2: code_integer,Nb: nat,B2: code_integer] :
      ( ( power_8256067586552552935nteger(A2,aa_nat_nat(suc,Nb)) = power_8256067586552552935nteger(B2,aa_nat_nat(suc,Nb)) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_250_power__inject__base,axiom,
    ! [A2: rat,Nb: nat,B2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B2),aa_nat_nat(suc,Nb)) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_251_power__inject__base,axiom,
    ! [A2: nat,Nb: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(suc,Nb)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_252_power__inject__base,axiom,
    ! [A2: int,Nb: nat,B2: int] :
      ( ( aa_nat_int(power_power_int(A2),aa_nat_nat(suc,Nb)) = aa_nat_int(power_power_int(B2),aa_nat_nat(suc,Nb)) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
tff(fact_253_div__greater__zero__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_254_div__le__mono2,axiom,
    ! [M: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),M)) ) ) ).

% div_le_mono2
tff(fact_255_nat__one__le__power,axiom,
    ! [I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb)) ) ).

% nat_one_le_power
tff(fact_256_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,V)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_257_power__eq__imp__eq__base,axiom,
    ! [A2: real,Nb: nat,B2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_258_power__eq__imp__eq__base,axiom,
    ! [A2: code_integer,Nb: nat,B2: code_integer] :
      ( ( power_8256067586552552935nteger(A2,Nb) = power_8256067586552552935nteger(B2,Nb) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_259_power__eq__imp__eq__base,axiom,
    ! [A2: rat,Nb: nat,B2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),Nb) = aa_nat_rat(power_power_rat(B2),Nb) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_260_power__eq__imp__eq__base,axiom,
    ! [A2: nat,Nb: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_261_power__eq__imp__eq__base,axiom,
    ! [A2: int,Nb: nat,B2: int] :
      ( ( aa_nat_int(power_power_int(A2),Nb) = aa_nat_int(power_power_int(B2),Nb) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_262_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A2: real,B2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_263_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A2: code_integer,B2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
         => ( ( power_8256067586552552935nteger(A2,Nb) = power_8256067586552552935nteger(B2,Nb) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_264_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A2: rat,B2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
         => ( ( aa_nat_rat(power_power_rat(A2),Nb) = aa_nat_rat(power_power_rat(B2),Nb) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_265_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_266_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A2: int,B2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
         => ( ( aa_nat_int(power_power_int(A2),Nb) = aa_nat_int(power_power_int(B2),Nb) )
          <=> ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_267_power2__nat__le__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_268_power2__nat__le__eq__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_269_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_270_div__nat__eqI,axiom,
    ! [Nb: nat,Q3: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q3)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_271_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Q3))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_272_is__num__normalize_I1_J,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% is_num_normalize(1)
tff(fact_273_is__num__normalize_I1_J,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) ).

% is_num_normalize(1)
tff(fact_274_is__num__normalize_I1_J,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A2),B2)),C2) = aa_int_int(plus_plus_int(A2),aa_int_int(plus_plus_int(B2),C2)) ).

% is_num_normalize(1)
tff(fact_275_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_276_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B2),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_277_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),aa_num_real(numeral_numeral_real,W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),B2),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_278_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),C2)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_279_power2__le__imp__le,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ) ).

% power2_le_imp_le
tff(fact_280_power2__le__imp__le,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ya)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xc),Ya) ) ) ).

% power2_le_imp_le
tff(fact_281_power2__le__imp__le,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya) ) ) ).

% power2_le_imp_le
tff(fact_282_power2__le__imp__le,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ya)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya) ) ) ).

% power2_le_imp_le
tff(fact_283_power2__le__imp__le,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya) ) ) ).

% power2_le_imp_le
tff(fact_284_power2__eq__imp__eq,axiom,
    ! [Xc: real,Ya: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
         => ( Xc = Ya ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_285_power2__eq__imp__eq,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( ( power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xc)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ya)
         => ( Xc = Ya ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_286_power2__eq__imp__eq,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ( aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
         => ( Xc = Ya ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_287_power2__eq__imp__eq,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xc)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ya)
         => ( Xc = Ya ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_288_power2__eq__imp__eq,axiom,
    ! [Xc: int,Ya: int] :
      ( ( aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
         => ( Xc = Ya ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_289_zero__le__power2,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_290_zero__le__power2,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_291_zero__le__power2,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_292_zero__le__power2,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% zero_le_power2
tff(fact_293_power__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(A2,Nb)),power_8256067586552552935nteger(B2,Nb)) ) ) ) ).

% power_strict_mono
tff(fact_294_power__strict__mono,axiom,
    ! [A2: real,B2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_295_power__strict__mono,axiom,
    ! [A2: rat,B2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),Nb)),aa_nat_rat(power_power_rat(B2),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_296_power__strict__mono,axiom,
    ! [A2: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_297_power__strict__mono,axiom,
    ! [A2: int,B2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(B2),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_298_split__div_H,axiom,
    ! [P: fun_nat_o,M: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q4)),M)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_299_power2__less__imp__less,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ya)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xc),Ya) ) ) ).

% power2_less_imp_less
tff(fact_300_power2__less__imp__less,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ) ).

% power2_less_imp_less
tff(fact_301_power2__less__imp__less,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya) ) ) ).

% power2_less_imp_less
tff(fact_302_power2__less__imp__less,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ya)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya) ) ) ).

% power2_less_imp_less
tff(fact_303_power2__less__imp__less,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya) ) ) ).

% power2_less_imp_less
tff(fact_304_sum__power2__le__zero__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( Xc = zero_zero_real )
        & ( Ya = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_305_sum__power2__le__zero__iff,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,plus_p5714425477246183910nteger(power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger)
    <=> ( ( Xc = zero_z3403309356797280102nteger )
        & ( Ya = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_306_sum__power2__le__zero__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( Xc = zero_zero_rat )
        & ( Ya = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_307_sum__power2__le__zero__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( Xc = zero_zero_int )
        & ( Ya = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_308_sum__power2__ge__zero,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_309_sum__power2__ge__zero,axiom,
    ! [Xc: code_integer,Ya: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_310_sum__power2__ge__zero,axiom,
    ! [Xc: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_311_sum__power2__ge__zero,axiom,
    ! [Xc: int,Ya: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_312_zero__le__even__power_H,axiom,
    ! [A2: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_313_zero__le__even__power_H,axiom,
    ! [A2: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_314_zero__le__even__power_H,axiom,
    ! [A2: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_315_zero__le__even__power_H,axiom,
    ! [A2: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% zero_le_even_power'
tff(fact_316_odd__0__le__power__imp__0__le,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% odd_0_le_power_imp_0_le
tff(fact_317_odd__0__le__power__imp__0__le,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ).

% odd_0_le_power_imp_0_le
tff(fact_318_odd__0__le__power__imp__0__le,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% odd_0_le_power_imp_0_le
tff(fact_319_odd__0__le__power__imp__0__le,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% odd_0_le_power_imp_0_le
tff(fact_320_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_321_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Nb) ).

% zero_neq_numeral
tff(fact_322_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Nb) ).

% zero_neq_numeral
tff(fact_323_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_324_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_325_less__numeral__extra_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_zero_real) ).

% less_numeral_extra(3)
tff(fact_326_less__numeral__extra_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_zero_rat) ).

% less_numeral_extra(3)
tff(fact_327_less__numeral__extra_I3_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_zero_nat) ).

% less_numeral_extra(3)
tff(fact_328_less__numeral__extra_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_numeral_extra(3)
tff(fact_329_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: rat,Nb: nat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_nat_rat(power_power_rat(A2),Nb) != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_330_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: nat,Nb: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb) != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_331_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: real,Nb: nat] :
      ( ( A2 != zero_zero_real )
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb) != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_332_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: int,Nb: nat] :
      ( ( A2 != zero_zero_int )
     => ( aa_nat_int(power_power_int(A2),Nb) != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_333_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: complex,Nb: nat] :
      ( ( A2 != zero_zero_complex )
     => ( aa_nat_complex(power_power_complex(A2),Nb) != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_334_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( power_8256067586552552935nteger(A2,Nb) != zero_z3403309356797280102nteger ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_335_power__commuting__commutes,axiom,
    ! [Xc: complex,Ya: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xc),Ya) = aa_complex_complex(times_times_complex(Ya),Xc) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xc),Nb)),Ya) = aa_complex_complex(times_times_complex(Ya),aa_nat_complex(power_power_complex(Xc),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_336_power__commuting__commutes,axiom,
    ! [Xc: code_integer,Ya: code_integer,Nb: nat] :
      ( ( times_3573771949741848930nteger(Xc,Ya) = times_3573771949741848930nteger(Ya,Xc) )
     => ( times_3573771949741848930nteger(power_8256067586552552935nteger(Xc,Nb),Ya) = times_3573771949741848930nteger(Ya,power_8256067586552552935nteger(Xc,Nb)) ) ) ).

% power_commuting_commutes
tff(fact_337_power__commuting__commutes,axiom,
    ! [Xc: real,Ya: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Xc) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_338_power__commuting__commutes,axiom,
    ! [Xc: rat,Ya: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xc),Ya) = aa_rat_rat(times_times_rat(Ya),Xc) )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xc),Nb)),Ya) = aa_rat_rat(times_times_rat(Ya),aa_nat_rat(power_power_rat(Xc),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_339_power__commuting__commutes,axiom,
    ! [Xc: nat,Ya: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xc),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ya),Xc) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),Nb)),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_340_power__commuting__commutes,axiom,
    ! [Xc: int,Ya: int,Nb: nat] :
      ( ( aa_int_int(times_times_int(Xc),Ya) = aa_int_int(times_times_int(Ya),Xc) )
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(Xc),Nb)),Ya) = aa_int_int(times_times_int(Ya),aa_nat_int(power_power_int(Xc),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_341_power__commuting__commutes,axiom,
    ! [Xc: assn,Ya: assn,Nb: nat] :
      ( ( aa_assn_assn(times_times_assn(Xc),Ya) = aa_assn_assn(times_times_assn(Ya),Xc) )
     => ( aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(Xc),Nb)),Ya) = aa_assn_assn(times_times_assn(Ya),aa_nat_assn(power_power_assn(Xc),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_342_power__mult__distrib,axiom,
    ! [A2: complex,B2: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A2),B2)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A2),Nb)),aa_nat_complex(power_power_complex(B2),Nb)) ).

% power_mult_distrib
tff(fact_343_power__mult__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,Nb: nat] : power_8256067586552552935nteger(times_3573771949741848930nteger(A2,B2),Nb) = times_3573771949741848930nteger(power_8256067586552552935nteger(A2,Nb),power_8256067586552552935nteger(B2,Nb)) ).

% power_mult_distrib
tff(fact_344_power__mult__distrib,axiom,
    ! [A2: real,B2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb)) ).

% power_mult_distrib
tff(fact_345_power__mult__distrib,axiom,
    ! [A2: rat,B2: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(times_times_rat(A2),B2)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),Nb)),aa_nat_rat(power_power_rat(B2),Nb)) ).

% power_mult_distrib
tff(fact_346_power__mult__distrib,axiom,
    ! [A2: nat,B2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb)) ).

% power_mult_distrib
tff(fact_347_power__mult__distrib,axiom,
    ! [A2: int,B2: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(times_times_int(A2),B2)),Nb) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(B2),Nb)) ).

% power_mult_distrib
tff(fact_348_power__mult__distrib,axiom,
    ! [A2: assn,B2: assn,Nb: nat] : aa_nat_assn(power_power_assn(aa_assn_assn(times_times_assn(A2),B2)),Nb) = aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),Nb)),aa_nat_assn(power_power_assn(B2),Nb)) ).

% power_mult_distrib
tff(fact_349_power__commutes,axiom,
    ! [A2: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A2),Nb)),A2) = aa_complex_complex(times_times_complex(A2),aa_nat_complex(power_power_complex(A2),Nb)) ).

% power_commutes
tff(fact_350_power__commutes,axiom,
    ! [A2: code_integer,Nb: nat] : times_3573771949741848930nteger(power_8256067586552552935nteger(A2,Nb),A2) = times_3573771949741848930nteger(A2,power_8256067586552552935nteger(A2,Nb)) ).

% power_commutes
tff(fact_351_power__commutes,axiom,
    ! [A2: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ).

% power_commutes
tff(fact_352_power__commutes,axiom,
    ! [A2: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),Nb)),A2) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),Nb)) ).

% power_commutes
tff(fact_353_power__commutes,axiom,
    ! [A2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ).

% power_commutes
tff(fact_354_power__commutes,axiom,
    ! [A2: int,Nb: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(A2),Nb)),A2) = aa_int_int(times_times_int(A2),aa_nat_int(power_power_int(A2),Nb)) ).

% power_commutes
tff(fact_355_power__commutes,axiom,
    ! [A2: assn,Nb: nat] : aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),Nb)),A2) = aa_assn_assn(times_times_assn(A2),aa_nat_assn(power_power_assn(A2),Nb)) ).

% power_commutes
tff(fact_356_power__divide,axiom,
    ! [A2: complex,B2: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A2),B2)),Nb) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A2),Nb)),aa_nat_complex(power_power_complex(B2),Nb)) ).

% power_divide
tff(fact_357_power__divide,axiom,
    ! [A2: real,B2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb)) ).

% power_divide
tff(fact_358_power__divide,axiom,
    ! [A2: rat,B2: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A2),B2)),Nb) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A2),Nb)),aa_nat_rat(power_power_rat(B2),Nb)) ).

% power_divide
tff(fact_359_power__mult,axiom,
    ! [A2: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),Nb) ).

% power_mult
tff(fact_360_power__mult,axiom,
    ! [A2: real,M: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),Nb) ).

% power_mult
tff(fact_361_power__mult,axiom,
    ! [A2: int,M: nat,Nb: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A2),M)),Nb) ).

% power_mult
tff(fact_362_power__mult,axiom,
    ! [A2: complex,M: nat,Nb: nat] : aa_nat_complex(power_power_complex(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A2),M)),Nb) ).

% power_mult
tff(fact_363_power__mult,axiom,
    ! [A2: code_integer,M: nat,Nb: nat] : power_8256067586552552935nteger(A2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = power_8256067586552552935nteger(power_8256067586552552935nteger(A2,M),Nb) ).

% power_mult
tff(fact_364_div__mult2__eq,axiom,
    ! [M: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),Q3) ).

% div_mult2_eq
tff(fact_365_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),zero_zero_real) ).

% not_numeral_less_zero
tff(fact_366_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),zero_zero_rat) ).

% not_numeral_less_zero
tff(fact_367_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),zero_zero_nat) ).

% not_numeral_less_zero
tff(fact_368_not__numeral__less__zero,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),zero_zero_int) ).

% not_numeral_less_zero
tff(fact_369_zero__less__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,Nb)) ).

% zero_less_numeral
tff(fact_370_zero__less__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_less_numeral
tff(fact_371_zero__less__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% zero_less_numeral
tff(fact_372_zero__less__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_num_int(numeral_numeral_int,Nb)) ).

% zero_less_numeral
tff(fact_373_numeral__Bit0,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb)) ).

% numeral_Bit0
tff(fact_374_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_Bit0
tff(fact_375_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_Bit0
tff(fact_376_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_Bit0
tff(fact_377_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_Bit0
tff(fact_378_mult__numeral__1__right,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(A2),numera6690914467698888265omplex(one)) = A2 ).

% mult_numeral_1_right
tff(fact_379_mult__numeral__1__right,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,one)) = A2 ).

% mult_numeral_1_right
tff(fact_380_mult__numeral__1__right,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),aa_num_rat(numeral_numeral_rat,one)) = A2 ).

% mult_numeral_1_right
tff(fact_381_mult__numeral__1__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_num_nat(numeral_numeral_nat,one)) = A2 ).

% mult_numeral_1_right
tff(fact_382_mult__numeral__1__right,axiom,
    ! [A2: int] : aa_int_int(times_times_int(A2),aa_num_int(numeral_numeral_int,one)) = A2 ).

% mult_numeral_1_right
tff(fact_383_mult__numeral__1,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A2) = A2 ).

% mult_numeral_1
tff(fact_384_mult__numeral__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,one)),A2) = A2 ).

% mult_numeral_1
tff(fact_385_mult__numeral__1,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,one)),A2) = A2 ).

% mult_numeral_1
tff(fact_386_mult__numeral__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,one)),A2) = A2 ).

% mult_numeral_1
tff(fact_387_mult__numeral__1,axiom,
    ! [A2: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,one)),A2) = A2 ).

% mult_numeral_1
tff(fact_388_zero__less__power,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,Nb)) ) ).

% zero_less_power
tff(fact_389_zero__less__power,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ).

% zero_less_power
tff(fact_390_zero__less__power,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),Nb)) ) ).

% zero_less_power
tff(fact_391_zero__less__power,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ) ).

% zero_less_power
tff(fact_392_zero__less__power,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A2),Nb)) ) ).

% zero_less_power
tff(fact_393_divide__numeral__1,axiom,
    ! [A2: complex] : aa_complex_complex(divide1717551699836669952omplex(A2),numera6690914467698888265omplex(one)) = A2 ).

% divide_numeral_1
tff(fact_394_divide__numeral__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_num_real(numeral_numeral_real,one)) = A2 ).

% divide_numeral_1
tff(fact_395_divide__numeral__1,axiom,
    ! [A2: rat] : aa_rat_rat(divide_divide_rat(A2),aa_num_rat(numeral_numeral_rat,one)) = A2 ).

% divide_numeral_1
tff(fact_396_power__Suc2,axiom,
    ! [A2: complex,Nb: nat] : aa_nat_complex(power_power_complex(A2),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A2),Nb)),A2) ).

% power_Suc2
tff(fact_397_power__Suc2,axiom,
    ! [A2: code_integer,Nb: nat] : power_8256067586552552935nteger(A2,aa_nat_nat(suc,Nb)) = times_3573771949741848930nteger(power_8256067586552552935nteger(A2,Nb),A2) ).

% power_Suc2
tff(fact_398_power__Suc2,axiom,
    ! [A2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),A2) ).

% power_Suc2
tff(fact_399_power__Suc2,axiom,
    ! [A2: rat,Nb: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),Nb)),A2) ).

% power_Suc2
tff(fact_400_power__Suc2,axiom,
    ! [A2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),A2) ).

% power_Suc2
tff(fact_401_power__Suc2,axiom,
    ! [A2: int,Nb: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(suc,Nb)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A2),Nb)),A2) ).

% power_Suc2
tff(fact_402_power__Suc2,axiom,
    ! [A2: assn,Nb: nat] : aa_nat_assn(power_power_assn(A2),aa_nat_nat(suc,Nb)) = aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),Nb)),A2) ).

% power_Suc2
tff(fact_403_power__Suc,axiom,
    ! [A2: complex,Nb: nat] : aa_nat_complex(power_power_complex(A2),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A2),aa_nat_complex(power_power_complex(A2),Nb)) ).

% power_Suc
tff(fact_404_power__Suc,axiom,
    ! [A2: code_integer,Nb: nat] : power_8256067586552552935nteger(A2,aa_nat_nat(suc,Nb)) = times_3573771949741848930nteger(A2,power_8256067586552552935nteger(A2,Nb)) ).

% power_Suc
tff(fact_405_power__Suc,axiom,
    ! [A2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ).

% power_Suc
tff(fact_406_power__Suc,axiom,
    ! [A2: rat,Nb: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,Nb)) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),Nb)) ).

% power_Suc
tff(fact_407_power__Suc,axiom,
    ! [A2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ).

% power_Suc
tff(fact_408_power__Suc,axiom,
    ! [A2: int,Nb: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(suc,Nb)) = aa_int_int(times_times_int(A2),aa_nat_int(power_power_int(A2),Nb)) ).

% power_Suc
tff(fact_409_power__Suc,axiom,
    ! [A2: assn,Nb: nat] : aa_nat_assn(power_power_assn(A2),aa_nat_nat(suc,Nb)) = aa_assn_assn(times_times_assn(A2),aa_nat_assn(power_power_assn(A2),Nb)) ).

% power_Suc
tff(fact_410_power__add,axiom,
    ! [A2: complex,M: nat,Nb: nat] : aa_nat_complex(power_power_complex(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A2),M)),aa_nat_complex(power_power_complex(A2),Nb)) ).

% power_add
tff(fact_411_power__add,axiom,
    ! [A2: code_integer,M: nat,Nb: nat] : power_8256067586552552935nteger(A2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = times_3573771949741848930nteger(power_8256067586552552935nteger(A2,M),power_8256067586552552935nteger(A2,Nb)) ).

% power_add
tff(fact_412_power__add,axiom,
    ! [A2: real,M: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ).

% power_add
tff(fact_413_power__add,axiom,
    ! [A2: rat,M: nat,Nb: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),Nb)) ).

% power_add
tff(fact_414_power__add,axiom,
    ! [A2: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ).

% power_add
tff(fact_415_power__add,axiom,
    ! [A2: int,M: nat,Nb: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_int_int(times_times_int(aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),Nb)) ).

% power_add
tff(fact_416_power__add,axiom,
    ! [A2: assn,M: nat,Nb: nat] : aa_nat_assn(power_power_assn(A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),M)),aa_nat_assn(power_power_assn(A2),Nb)) ).

% power_add
tff(fact_417_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_418_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_419_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_420_sum__squares__gt__zero__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya)))
    <=> ( ( Xc != zero_zero_real )
        | ( Ya != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_421_sum__squares__gt__zero__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xc),Xc)),aa_rat_rat(times_times_rat(Ya),Ya)))
    <=> ( ( Xc != zero_zero_rat )
        | ( Ya != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_422_sum__squares__gt__zero__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xc),Xc)),aa_int_int(times_times_int(Ya),Ya)))
    <=> ( ( Xc != zero_zero_int )
        | ( Ya != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_423_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: complex,C2: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C2) = B2,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_424_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( ( aa_num_real(numeral_numeral_real,W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
    <=> $ite(C2 != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2) = B2,aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_425_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(divide_divide_rat(B2),C2) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2) = B2,aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_426_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: complex,C2: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C2) = numera6690914467698888265omplex(W) )
    <=> $ite(C2 != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C2),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_427_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = aa_num_real(numeral_numeral_real,W) )
    <=> $ite(C2 != zero_zero_real,B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2),aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% divide_eq_eq_numeral(1)
tff(fact_428_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B2),C2) = aa_num_rat(numeral_numeral_rat,W) )
    <=> $ite(C2 != zero_zero_rat,B2 = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2),aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_429_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit0_div_2
tff(fact_430_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit0_div_2
tff(fact_431_numeral__1__eq__Suc__0,axiom,
    aa_num_nat(numeral_numeral_nat,one) = aa_nat_nat(suc,zero_zero_nat) ).

% numeral_1_eq_Suc_0
tff(fact_432_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = zero_zero_rat ) ) ).

% zero_power
tff(fact_433_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = zero_zero_nat ) ) ).

% zero_power
tff(fact_434_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = zero_zero_real ) ) ).

% zero_power
tff(fact_435_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(power_power_int(zero_zero_int),Nb) = zero_zero_int ) ) ).

% zero_power
tff(fact_436_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = zero_zero_complex ) ) ).

% zero_power
tff(fact_437_zero__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( power_8256067586552552935nteger(zero_z3403309356797280102nteger,Nb) = zero_z3403309356797280102nteger ) ) ).

% zero_power
tff(fact_438_power__gt__expt,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),K)) ) ).

% power_gt_expt
tff(fact_439_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Q3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Q3)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).

% div_less_iff_less_mult
tff(fact_440_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_441_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(divide_divide_rat(B2),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_442_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),aa_num_real(numeral_numeral_real,W))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C2)),B2),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_num_real(numeral_numeral_real,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_443_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),C2)),aa_num_rat(numeral_numeral_rat,W))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),C2)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_num_rat(numeral_numeral_rat,W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_444_left__add__twice,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(plus_plus_complex(A2),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_445_left__add__twice,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_446_left__add__twice,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(plus_plus_rat(A2),B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_447_left__add__twice,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_448_left__add__twice,axiom,
    ! [A2: int,B2: int] : aa_int_int(plus_plus_int(A2),aa_int_int(plus_plus_int(A2),B2)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),B2) ).

% left_add_twice
tff(fact_449_mult__2__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2_right
tff(fact_450_mult__2__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2_right
tff(fact_451_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2_right
tff(fact_452_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_453_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(times_times_int(Z),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(plus_plus_int(Z),Z) ).

% mult_2_right
tff(fact_454_mult__2,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Z) = aa_complex_complex(plus_plus_complex(Z),Z) ).

% mult_2
tff(fact_455_mult__2,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Z) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),Z) ).

% mult_2
tff(fact_456_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Z) = aa_rat_rat(plus_plus_rat(Z),Z) ).

% mult_2
tff(fact_457_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_458_mult__2,axiom,
    ! [Z: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Z) = aa_int_int(plus_plus_int(Z),Z) ).

% mult_2
tff(fact_459_zero__power2,axiom,
    aa_nat_rat(power_power_rat(zero_zero_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_rat ).

% zero_power2
tff(fact_460_zero__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% zero_power2
tff(fact_461_zero__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_real ).

% zero_power2
tff(fact_462_zero__power2,axiom,
    aa_nat_int(power_power_int(zero_zero_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_int ).

% zero_power2
tff(fact_463_zero__power2,axiom,
    aa_nat_complex(power_power_complex(zero_zero_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_complex ).

% zero_power2
tff(fact_464_zero__power2,axiom,
    power_8256067586552552935nteger(zero_z3403309356797280102nteger,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% zero_power2
tff(fact_465_power2__eq__square,axiom,
    ! [A2: complex] : aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(times_times_complex(A2),A2) ).

% power2_eq_square
tff(fact_466_power2__eq__square,axiom,
    ! [A2: code_integer] : power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = times_3573771949741848930nteger(A2,A2) ).

% power2_eq_square
tff(fact_467_power2__eq__square,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2) ).

% power2_eq_square
tff(fact_468_power2__eq__square,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(A2),A2) ).

% power2_eq_square
tff(fact_469_power2__eq__square,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),A2) ).

% power2_eq_square
tff(fact_470_power2__eq__square,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(times_times_int(A2),A2) ).

% power2_eq_square
tff(fact_471_power2__eq__square,axiom,
    ! [A2: assn] : aa_nat_assn(power_power_assn(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_assn_assn(times_times_assn(A2),A2) ).

% power2_eq_square
tff(fact_472_power4__eq__xxxx,axiom,
    ! [Xc: complex] : aa_nat_complex(power_power_complex(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Xc),Xc)),Xc)),Xc) ).

% power4_eq_xxxx
tff(fact_473_power4__eq__xxxx,axiom,
    ! [Xc: code_integer] : power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = times_3573771949741848930nteger(times_3573771949741848930nteger(times_3573771949741848930nteger(Xc,Xc),Xc),Xc) ).

% power4_eq_xxxx
tff(fact_474_power4__eq__xxxx,axiom,
    ! [Xc: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc)),Xc)),Xc) ).

% power4_eq_xxxx
tff(fact_475_power4__eq__xxxx,axiom,
    ! [Xc: rat] : aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Xc),Xc)),Xc)),Xc) ).

% power4_eq_xxxx
tff(fact_476_power4__eq__xxxx,axiom,
    ! [Xc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xc),Xc)),Xc)),Xc) ).

% power4_eq_xxxx
tff(fact_477_power4__eq__xxxx,axiom,
    ! [Xc: int] : aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(times_times_int(Xc),Xc)),Xc)),Xc) ).

% power4_eq_xxxx
tff(fact_478_power4__eq__xxxx,axiom,
    ! [Xc: assn] : aa_nat_assn(power_power_assn(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(Xc),Xc)),Xc)),Xc) ).

% power4_eq_xxxx
tff(fact_479_numeral__2__eq__2,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% numeral_2_eq_2
tff(fact_480_power2__commute,axiom,
    ! [Xc: complex,Ya: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Ya),Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_481_power2__commute,axiom,
    ! [Xc: code_integer,Ya: code_integer] : power_8256067586552552935nteger(minus_8373710615458151222nteger(Xc,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = power_8256067586552552935nteger(minus_8373710615458151222nteger(Ya,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_482_power2__commute,axiom,
    ! [Xc: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_483_power2__commute,axiom,
    ! [Xc: rat,Ya: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Ya),Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_484_power2__commute,axiom,
    ! [Xc: int,Ya: int] : aa_nat_int(power_power_int(aa_int_int(minus_minus_int(Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(aa_int_int(minus_minus_int(Ya),Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_485_power__even__eq,axiom,
    ! [A2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_486_power__even__eq,axiom,
    ! [A2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_487_power__even__eq,axiom,
    ! [A2: int,Nb: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A2),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_488_power__even__eq,axiom,
    ! [A2: complex,Nb: nat] : aa_nat_complex(power_power_complex(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A2),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_489_power__even__eq,axiom,
    ! [A2: code_integer,Nb: nat] : power_8256067586552552935nteger(A2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = power_8256067586552552935nteger(power_8256067586552552935nteger(A2,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power_even_eq
tff(fact_490_dividend__less__times__div,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)))) ) ).

% dividend_less_times_div
tff(fact_491_dividend__less__div__times,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_492_split__div,axiom,
    ! [P: fun_nat_o,M: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I2: nat,J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),Nb)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I2)),J) )
               => aa_nat_o(P,I2) ) ) ) ) ) ).

% split_div
tff(fact_493_half__gt__zero__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ).

% half_gt_zero_iff
tff(fact_494_half__gt__zero__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% half_gt_zero_iff
tff(fact_495_half__gt__zero,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_496_half__gt__zero,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A2),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% half_gt_zero
tff(fact_497_power2__less__0,axiom,
    ! [A2: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_z3403309356797280102nteger) ).

% power2_less_0
tff(fact_498_power2__less__0,axiom,
    ! [A2: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_real) ).

% power2_less_0
tff(fact_499_power2__less__0,axiom,
    ! [A2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_rat) ).

% power2_less_0
tff(fact_500_power2__less__0,axiom,
    ! [A2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),zero_zero_int) ).

% power2_less_0
tff(fact_501_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
    <=> ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases_iff
tff(fact_502_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))
     => ( ( Nb = zero_zero_nat )
        | ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% less_2_cases
tff(fact_503_sum__power2__gt__zero__iff,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xc != zero_z3403309356797280102nteger )
        | ( Ya != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_504_sum__power2__gt__zero__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xc != zero_zero_real )
        | ( Ya != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_505_sum__power2__gt__zero__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xc != zero_zero_rat )
        | ( Ya != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_506_sum__power2__gt__zero__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xc != zero_zero_int )
        | ( Ya != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_507_not__sum__power2__lt__zero,axiom,
    ! [Xc: code_integer,Ya: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,plus_p5714425477246183910nteger(power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_z3403309356797280102nteger) ).

% not_sum_power2_lt_zero
tff(fact_508_not__sum__power2__lt__zero,axiom,
    ! [Xc: real,Ya: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_509_not__sum__power2__lt__zero,axiom,
    ! [Xc: rat,Ya: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_510_not__sum__power2__lt__zero,axiom,
    ! [Xc: int,Ya: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_511_power2__sum,axiom,
    ! [Xc: code_integer,Ya: code_integer] : power_8256067586552552935nteger(plus_p5714425477246183910nteger(Xc,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),times_3573771949741848930nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Xc),Ya)) ).

% power2_sum
tff(fact_512_power2__sum,axiom,
    ! [Xc: complex,Ya: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xc)),Ya)) ).

% power2_sum
tff(fact_513_power2__sum,axiom,
    ! [Xc: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xc)),Ya)) ).

% power2_sum
tff(fact_514_power2__sum,axiom,
    ! [Xc: rat,Ya: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(plus_plus_rat(Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xc)),Ya)) ).

% power2_sum
tff(fact_515_power2__sum,axiom,
    ! [Xc: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xc)),Ya)) ).

% power2_sum
tff(fact_516_power2__sum,axiom,
    ! [Xc: int,Ya: int] : aa_nat_int(power_power_int(aa_int_int(plus_plus_int(Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(times_times_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xc)),Ya)) ).

% power2_sum
tff(fact_517_power__odd__eq,axiom,
    ! [A2: complex,Nb: nat] : aa_nat_complex(power_power_complex(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(times_times_complex(A2),aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A2),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_518_power__odd__eq,axiom,
    ! [A2: code_integer,Nb: nat] : power_8256067586552552935nteger(A2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = times_3573771949741848930nteger(A2,power_8256067586552552935nteger(power_8256067586552552935nteger(A2,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_519_power__odd__eq,axiom,
    ! [A2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_520_power__odd__eq,axiom,
    ! [A2: rat,Nb: nat] : aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A2),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_521_power__odd__eq,axiom,
    ! [A2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_522_power__odd__eq,axiom,
    ! [A2: int,Nb: nat] : aa_nat_int(power_power_int(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(times_times_int(A2),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A2),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_523_power__odd__eq,axiom,
    ! [A2: assn,Nb: nat] : aa_nat_assn(power_power_assn(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_assn_assn(times_times_assn(A2),aa_nat_assn(power_power_assn(aa_nat_assn(power_power_assn(A2),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_524_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_525_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% div_2_gt_zero
tff(fact_526_power2__diff,axiom,
    ! [Xc: code_integer,Ya: code_integer] : power_8256067586552552935nteger(minus_8373710615458151222nteger(Xc,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),times_3573771949741848930nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Xc),Ya)) ).

% power2_diff
tff(fact_527_power2__diff,axiom,
    ! [Xc: complex,Ya: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xc)),Ya)) ).

% power2_diff
tff(fact_528_power2__diff,axiom,
    ! [Xc: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xc)),Ya)) ).

% power2_diff
tff(fact_529_power2__diff,axiom,
    ! [Xc: rat,Ya: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xc)),Ya)) ).

% power2_diff
tff(fact_530_power2__diff,axiom,
    ! [Xc: int,Ya: int] : aa_nat_int(power_power_int(aa_int_int(minus_minus_int(Xc),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(times_times_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xc)),Ya)) ).

% power2_diff
tff(fact_531_odd__power__less__zero,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(A2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_z3403309356797280102nteger) ) ).

% odd_power_less_zero
tff(fact_532_odd__power__less__zero,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_real) ) ).

% odd_power_less_zero
tff(fact_533_odd__power__less__zero,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_rat) ) ).

% odd_power_less_zero
tff(fact_534_odd__power__less__zero,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_zero_int) ) ).

% odd_power_less_zero
tff(fact_535_highboundn,axiom,
    ( ( ma != mi )
   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,xa),ma)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xnew),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(treeList)) ) ) ).

% highboundn
tff(fact_536_highbound,axiom,
    ( ( ma != mi )
   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,xa),ma)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(treeList)) ) ) ).

% highbound
tff(fact_537_xbound,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),xa)
   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,xa),ma)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(treeList)) ) ) ).

% xbound
tff(fact_538_big__assn__simp_H,axiom,
    ! [H: nat,TreeLista: list_VEBT_VEBT,Xaa: vEBT_VEBT,L: nat,Xc: vEBT_VEBTi,Xba: option_nat,X13a: array_VEBT_VEBTi,Tree_isa: list_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeLista))
     => ( ( Xaa = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L) )
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xaa),Xc)),aa_o_assn(pure_assn,Xba = vEBT_vebt_mint(Xaa)))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,H,Xc))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_isa))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,H,Xc))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_o_assn(pure_assn,Xba = vEBT_vebt_mint(Xaa)))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,H,Xaa)),list_u6098035379799741383_VEBTi(Tree_isa,H,Xc)))) ) ) ).

% big_assn_simp'
tff(fact_539_big__assn__simp,axiom,
    ! [H: nat,TreeLista: list_VEBT_VEBT,L: nat,Xc: vEBT_VEBTi,Xaa: option_nat,X13a: array_VEBT_VEBTi,Tree_isa: list_VEBT_VEBTi,Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,H),size_s6755466524823107622T_VEBT(TreeLista))
     => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L)),Xc)),aa_o_assn(pure_assn,Xaa = vEBT_vebt_mint(aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L))))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,H,Xc))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(TreeLista)),insert_nat(H,bot_bot_set_nat)),vEBT_vebt_assn_raw,TreeLista),Tree_isa))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(X13a),list_u6098035379799741383_VEBTi(Tree_isa,H,Xc))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),X14a))),aa_o_assn(pure_assn,Xaa = vEBT_vebt_mint(aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L))))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(TreeLista,H,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L))),list_u6098035379799741383_VEBTi(Tree_isa,H,Xc)))) ) ).

% big_assn_simp
tff(fact_540_mimaxprop,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma)
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)) ) ).

% mimaxprop
tff(fact_541_atLeastLessThan__singleton,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,M)) = insert_nat(M,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_542_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBT,J2: nat,Xc: vEBT_VEBT] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,J2,Xc)),I) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I) ) ) ).

% nth_update_invalid
tff(fact_543_nth__update__invalid,axiom,
    ! [I: nat,L: list_real,J2: nat,Xc: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_nat_real(nth_real(list_update_real(L,J2,Xc)),I) = aa_nat_real(nth_real(L),I) ) ) ).

% nth_update_invalid
tff(fact_544_nth__update__invalid,axiom,
    ! [I: nat,L: list_o,J2: nat,Xc: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( aa_nat_o(nth_o(list_update_o(L,J2,(Xc))),I)
      <=> aa_nat_o(nth_o(L),I) ) ) ).

% nth_update_invalid
tff(fact_545_nth__update__invalid,axiom,
    ! [I: nat,L: list_nat,J2: nat,Xc: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_nat_nat(nth_nat(list_update_nat(L,J2,Xc)),I) = aa_nat_nat(nth_nat(L),I) ) ) ).

% nth_update_invalid
tff(fact_546_nth__update__invalid,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,J2: nat,Xc: vEBT_VEBTi] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,J2,Xc),I) = nth_VEBT_VEBTi(L,I) ) ) ).

% nth_update_invalid
tff(fact_547_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xc: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xc)),I) = Xc ) ) ).

% nth_list_update_eq
tff(fact_548_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_real,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,Xc)),I) = Xc ) ) ).

% nth_list_update_eq
tff(fact_549_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,Xc: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xc))),I)
      <=> (Xc) ) ) ).

% nth_list_update_eq
tff(fact_550_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xc)),I) = Xc ) ) ).

% nth_list_update_eq
tff(fact_551_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xc: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xc),I) = Xc ) ) ).

% nth_list_update_eq
tff(fact_552_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_VE1571388752190420990T_assn,Xc: vEBT_VEBT,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,Xc),Xi)),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I3,A,list_u1324408373059187874T_VEBT(Xs,I,Xc),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_553_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_re560840401387337987i_assn,Xc: real,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,Xc),Xi)),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I3,A,list_update_real(Xs,I,Xc)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_554_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_re8050108611700907676T_assn,Xc: real,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,Xc),Xi)),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I3,A,list_update_real(Xs,I,Xc),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_555_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_o_9029690514724572677i_assn,Xc: $o,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBTi,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A,(Xc)),Xi)),aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(I3,A,list_update_o(Xs,I,(Xc))),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_556_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_o_4353168661238401050T_assn,Xc: $o,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBT,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A,(Xc)),Xi)),vEBT_L1319876754960170684T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L1319876754960170684T_VEBT(I3,A,list_update_o(Xs,I,(Xc)),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_557_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_na6467670216208870567i_assn,Xc: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A,Xc),Xi)),aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(I3,A,list_update_nat(Xs,I,Xc)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_558_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_na4489259242115493880T_assn,Xc: nat,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A,Xc),Xi)),vEBT_L8511957252848910786T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L8511957252848910786T_VEBT(I3,A,list_update_nat(Xs,I,Xc),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_559_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_VE2561185598974572172i_assn,Xc: vEBT_VEBTi,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A,Xc),Xi)),aa_lis2508312872377092907i_assn(vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L886525131989349516_VEBTi(I3,A,list_u6098035379799741383_VEBTi(Xs,I,Xc)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_560_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_VE5577048379444288083T_assn,Xc: vEBT_VEBTi,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A,Xc),Xi)),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L2497118539674116125T_VEBT(I3,A,list_u6098035379799741383_VEBTi(Xs,I,Xc),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_561_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: fun_VE905575822219544865i_assn,Xc: vEBT_VEBT,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(A,Xc),Xi)),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I3,A,list_u1324408373059187874T_VEBT(Xs,I,Xc)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_562_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_563_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% mult_le_cancel2
tff(fact_564_semiring__norm_I87_J,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_num(bit0,M) = aa_num_num(bit0,Nb) )
    <=> ( M = Nb ) ) ).

% semiring_norm(87)
tff(fact_565_diff__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ).

% diff_Suc_Suc
tff(fact_566_Suc__diff__diff,axiom,
    ! [M: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),K) ).

% Suc_diff_diff
tff(fact_567_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X22) = aa_nat_nat(suc,Y2) )
    <=> ( X22 = Y2 ) ) ).

% nat.inject
tff(fact_568_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_569_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% diff_0_eq_0
tff(fact_570_diff__self__eq__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_571_diff__diff__left,axiom,
    ! [I: nat,J2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K)) ).

% diff_diff_left
tff(fact_572_diff__diff__cancel,axiom,
    ! [I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_573_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xc: vEBT_VEBTi,Ya: vEBT_VEBTi] : list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xc),I,Ya) = list_u6098035379799741383_VEBTi(Xs,I,Ya) ).

% list_update_overwrite
tff(fact_574_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xc: vEBT_VEBT,Ya: vEBT_VEBT] : list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xc),I,Ya) = list_u1324408373059187874T_VEBT(Xs,I,Ya) ).

% list_update_overwrite
tff(fact_575_semiring__norm_I85_J,axiom,
    ! [M: num] : aa_num_num(bit0,M) != one ).

% semiring_norm(85)
tff(fact_576_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit0,Nb) ).

% semiring_norm(83)
tff(fact_577_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_578_Suc__mono,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_579_Suc__less__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_less_eq
tff(fact_580_zero__less__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% zero_less_diff
tff(fact_581_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_582_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% neq0_conv
tff(fact_583_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_nat_zero_code
tff(fact_584_add__Suc__right,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ).

% add_Suc_right
tff(fact_585_Suc__le__mono,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ).

% Suc_le_mono
tff(fact_586_add__is__0,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_587_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).

% Nat.add_0_right
tff(fact_588_diff__is__0__eq_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_589_diff__is__0__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% diff_is_0_eq
tff(fact_590_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2) ).

% bot_nat_0.extremum
tff(fact_591_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_592_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_593_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J2) ) ) ).

% Nat.diff_diff_right
tff(fact_594_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_595_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_596_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_597_mult__is__0,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = zero_zero_nat )
    <=> ( ( M = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_598_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_599_mult__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( M = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_600_mult__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K) )
    <=> ( ( M = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_601_ivl__subset,axiom,
    ! [I: rat,J2: rat,M: rat,Nb: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or4029947393144176647an_rat(I,J2)),set_or4029947393144176647an_rat(M,Nb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,J2),I)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),I)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,J2),Nb) ) ) ) ).

% ivl_subset
tff(fact_602_ivl__subset,axiom,
    ! [I: num,J2: num,M: num,Nb: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or1222409239386451017an_num(I,J2)),set_or1222409239386451017an_num(M,Nb))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,J2),I)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),I)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,J2),Nb) ) ) ) ).

% ivl_subset
tff(fact_603_ivl__subset,axiom,
    ! [I: nat,J2: nat,M: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),J2)),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),Nb) ) ) ) ).

% ivl_subset
tff(fact_604_ivl__subset,axiom,
    ! [I: int,J2: int,M: int,Nb: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or4662586982721622107an_int(I,J2)),set_or4662586982721622107an_int(M,Nb))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),I)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J2),Nb) ) ) ) ).

% ivl_subset
tff(fact_605_ivl__subset,axiom,
    ! [I: code_integer,J2: code_integer,M: code_integer,Nb: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(I,J2)),set_or8404916559141939852nteger(M,Nb))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,J2),I)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,M),I)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,J2),Nb) ) ) ) ).

% ivl_subset
tff(fact_606_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xc: vEBT_VEBT] : size_s6755466524823107622T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xc)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_list_update
tff(fact_607_length__list__update,axiom,
    ! [Xs: list_real,I: nat,Xc: real] : size_size_list_real(list_update_real(Xs,I,Xc)) = size_size_list_real(Xs) ).

% length_list_update
tff(fact_608_length__list__update,axiom,
    ! [Xs: list_o,I: nat,Xc: $o] : size_size_list_o(list_update_o(Xs,I,(Xc))) = size_size_list_o(Xs) ).

% length_list_update
tff(fact_609_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,Xc: nat] : size_size_list_nat(list_update_nat(Xs,I,Xc)) = size_size_list_nat(Xs) ).

% length_list_update
tff(fact_610_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xc: vEBT_VEBTi] : size_s7982070591426661849_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xc)) = size_s7982070591426661849_VEBTi(Xs) ).

% length_list_update
tff(fact_611_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] : list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),I)) = Xs ).

% list_update_id
tff(fact_612_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat] : list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,I)) = Xs ).

% list_update_id
tff(fact_613_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] : list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)) = Xs ).

% list_update_id
tff(fact_614_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs: list_nat,Xc: nat] :
      ( ( I != J2 )
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xc)),J2) = aa_nat_nat(nth_nat(Xs),J2) ) ) ).

% nth_list_update_neq
tff(fact_615_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs: list_VEBT_VEBTi,Xc: vEBT_VEBTi] :
      ( ( I != J2 )
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xc),J2) = nth_VEBT_VEBTi(Xs,J2) ) ) ).

% nth_list_update_neq
tff(fact_616_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs: list_VEBT_VEBT,Xc: vEBT_VEBT] :
      ( ( I != J2 )
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xc)),J2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J2) ) ) ).

% nth_list_update_neq
tff(fact_617_semiring__norm_I6_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(M),Nb)) ).

% semiring_norm(6)
tff(fact_618_semiring__norm_I13_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,M)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(times_times_num(M),Nb))) ).

% semiring_norm(13)
tff(fact_619_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa_num_num(times_times_num(one),Nb) = Nb ).

% semiring_norm(12)
tff(fact_620_semiring__norm_I11_J,axiom,
    ! [M: num] : aa_num_num(times_times_num(M),one) = M ).

% semiring_norm(11)
tff(fact_621_semiring__norm_I71_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Nb) ) ).

% semiring_norm(71)
tff(fact_622_semiring__norm_I78_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb) ) ).

% semiring_norm(78)
tff(fact_623_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,one),Nb) ).

% semiring_norm(68)
tff(fact_624_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,M),one) ).

% semiring_norm(75)
tff(fact_625_listlength,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% listlength
tff(fact_626_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ).

% zero_comp_diff_simps(1)
tff(fact_627_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ).

% zero_comp_diff_simps(1)
tff(fact_628_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ).

% zero_comp_diff_simps(1)
tff(fact_629_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ).

% zero_comp_diff_simps(2)
tff(fact_630_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ).

% zero_comp_diff_simps(2)
tff(fact_631_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(minus_minus_int(A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ).

% zero_comp_diff_simps(2)
tff(fact_632_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% Suc_pred
tff(fact_633_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% less_Suc0
tff(fact_634_zero__less__Suc,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(suc,Nb)) ).

% zero_less_Suc
tff(fact_635_atLeastLessThan__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( aa_set_real_o(member_real(I),set_or66887138388493659n_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_636_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or3540276404033026485et_nat(L,U))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(L),I)
        & aa_set_nat_o(ord_less_set_nat(I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_637_atLeastLessThan__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( aa_set_rat_o(member_rat(I),set_or4029947393144176647an_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_638_atLeastLessThan__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( aa_set_num_o(member_num(I),set_or1222409239386451017an_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_639_atLeastLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_or4665077453230672383an_nat(L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_640_atLeastLessThan__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I),set_or4662586982721622107an_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_641_atLeastLessThan__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I),set_or8404916559141939852nteger(L,U))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),I)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,I),U) ) ) ).

% atLeastLessThan_iff
tff(fact_642_add__gr__0,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_643_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J2)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_644_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_645_mult__eq__1__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_646_one__eq__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) )
    <=> ( ( M = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_647_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% mult_less_cancel2
tff(fact_648_nat__0__less__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_649_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_650_mult__Suc__right,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) ).

% mult_Suc_right
tff(fact_651_atLeastLessThan__empty,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( set_or66887138388493659n_real(A2,B2) = bot_bot_set_real ) ) ).

% atLeastLessThan_empty
tff(fact_652_atLeastLessThan__empty,axiom,
    ! [B2: $o,A2: $o] :
      ( aa_o_o(ord_less_eq_o((B2)),(A2))
     => ( set_or7139685690850216873Than_o((A2),(B2)) = bot_bot_set_o ) ) ).

% atLeastLessThan_empty
tff(fact_653_atLeastLessThan__empty,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
     => ( set_or3540276404033026485et_nat(A2,B2) = bot_bot_set_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_654_atLeastLessThan__empty,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( set_or4029947393144176647an_rat(A2,B2) = bot_bot_set_rat ) ) ).

% atLeastLessThan_empty
tff(fact_655_atLeastLessThan__empty,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
     => ( set_or1222409239386451017an_num(A2,B2) = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
tff(fact_656_atLeastLessThan__empty,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(A2),B2) = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
tff(fact_657_atLeastLessThan__empty,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( set_or4662586982721622107an_int(A2,B2) = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
tff(fact_658_atLeastLessThan__empty,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),A2)
     => ( set_or8404916559141939852nteger(A2,B2) = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
tff(fact_659_atLeastLessThan__empty__iff,axiom,
    ! [A2: $o,B2: $o] :
      ( ( set_or7139685690850216873Than_o((A2),(B2)) = bot_bot_set_o )
    <=> ~ aa_o_o(ord_less_o((A2)),(B2)) ) ).

% atLeastLessThan_empty_iff
tff(fact_660_atLeastLessThan__empty__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( set_or66887138388493659n_real(A2,B2) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% atLeastLessThan_empty_iff
tff(fact_661_atLeastLessThan__empty__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( set_or4029947393144176647an_rat(A2,B2) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% atLeastLessThan_empty_iff
tff(fact_662_atLeastLessThan__empty__iff,axiom,
    ! [A2: num,B2: num] :
      ( ( set_or1222409239386451017an_num(A2,B2) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ).

% atLeastLessThan_empty_iff
tff(fact_663_atLeastLessThan__empty__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A2),B2) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% atLeastLessThan_empty_iff
tff(fact_664_atLeastLessThan__empty__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( set_or4662586982721622107an_int(A2,B2) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% atLeastLessThan_empty_iff
tff(fact_665_atLeastLessThan__empty__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( set_or8404916559141939852nteger(A2,B2) = bot_bo3990330152332043303nteger )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2) ) ).

% atLeastLessThan_empty_iff
tff(fact_666_atLeastLessThan__empty__iff2,axiom,
    ! [A2: $o,B2: $o] :
      ( ( bot_bot_set_o = set_or7139685690850216873Than_o((A2),(B2)) )
    <=> ~ aa_o_o(ord_less_o((A2)),(B2)) ) ).

% atLeastLessThan_empty_iff2
tff(fact_667_atLeastLessThan__empty__iff2,axiom,
    ! [A2: real,B2: real] :
      ( ( bot_bot_set_real = set_or66887138388493659n_real(A2,B2) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% atLeastLessThan_empty_iff2
tff(fact_668_atLeastLessThan__empty__iff2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( bot_bot_set_rat = set_or4029947393144176647an_rat(A2,B2) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% atLeastLessThan_empty_iff2
tff(fact_669_atLeastLessThan__empty__iff2,axiom,
    ! [A2: num,B2: num] :
      ( ( bot_bot_set_num = set_or1222409239386451017an_num(A2,B2) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ).

% atLeastLessThan_empty_iff2
tff(fact_670_atLeastLessThan__empty__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or4665077453230672383an_nat(A2),B2) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% atLeastLessThan_empty_iff2
tff(fact_671_atLeastLessThan__empty__iff2,axiom,
    ! [A2: int,B2: int] :
      ( ( bot_bot_set_int = set_or4662586982721622107an_int(A2,B2) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% atLeastLessThan_empty_iff2
tff(fact_672_atLeastLessThan__empty__iff2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or8404916559141939852nteger(A2,B2) )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2) ) ).

% atLeastLessThan_empty_iff2
tff(fact_673_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = $ite(K = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_674_ivl__diff,axiom,
    ! [I: rat,Nb: rat,M: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),Nb)
     => ( minus_minus_set_rat(set_or4029947393144176647an_rat(I,M),set_or4029947393144176647an_rat(I,Nb)) = set_or4029947393144176647an_rat(Nb,M) ) ) ).

% ivl_diff
tff(fact_675_ivl__diff,axiom,
    ! [I: num,Nb: num,M: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),Nb)
     => ( minus_minus_set_num(set_or1222409239386451017an_num(I,M),set_or1222409239386451017an_num(I,Nb)) = set_or1222409239386451017an_num(Nb,M) ) ) ).

% ivl_diff
tff(fact_676_ivl__diff,axiom,
    ! [I: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
     => ( minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),M),aa_nat_set_nat(set_or4665077453230672383an_nat(I),Nb)) = aa_nat_set_nat(set_or4665077453230672383an_nat(Nb),M) ) ) ).

% ivl_diff
tff(fact_677_ivl__diff,axiom,
    ! [I: int,Nb: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),Nb)
     => ( minus_minus_set_int(set_or4662586982721622107an_int(I,M),set_or4662586982721622107an_int(I,Nb)) = set_or4662586982721622107an_int(Nb,M) ) ) ).

% ivl_diff
tff(fact_678_ivl__diff,axiom,
    ! [I: code_integer,Nb: code_integer,M: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I),Nb)
     => ( minus_2355218937544613996nteger(set_or8404916559141939852nteger(I,M),set_or8404916559141939852nteger(I,Nb)) = set_or8404916559141939852nteger(Nb,M) ) ) ).

% ivl_diff
tff(fact_679_semiring__norm_I2_J,axiom,
    aa_num_num(plus_plus_num(one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_680_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xc: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Xs)),I)
     => ( list_u1324408373059187874T_VEBT(Xs,I,Xc) = Xs ) ) ).

% list_update_beyond
tff(fact_681_list__update__beyond,axiom,
    ! [Xs: list_real,I: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Xs)),I)
     => ( list_update_real(Xs,I,Xc) = Xs ) ) ).

% list_update_beyond
tff(fact_682_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,Xc: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Xs)),I)
     => ( list_update_o(Xs,I,(Xc)) = Xs ) ) ).

% list_update_beyond
tff(fact_683_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),I)
     => ( list_update_nat(Xs,I,Xc) = Xs ) ) ).

% list_update_beyond
tff(fact_684_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xc: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s7982070591426661849_VEBTi(Xs)),I)
     => ( list_u6098035379799741383_VEBTi(Xs,I,Xc) = Xs ) ) ).

% list_update_beyond
tff(fact_685_num__double,axiom,
    ! [Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,one)),Nb) = aa_num_num(bit0,Nb) ).

% num_double
tff(fact_686_power__mult__numeral,axiom,
    ! [A2: nat,M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Nb))) ).

% power_mult_numeral
tff(fact_687_power__mult__numeral,axiom,
    ! [A2: real,M: num,Nb: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Nb))) ).

% power_mult_numeral
tff(fact_688_power__mult__numeral,axiom,
    ! [A2: int,M: num,Nb: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Nb))) ).

% power_mult_numeral
tff(fact_689_power__mult__numeral,axiom,
    ! [A2: complex,M: num,Nb: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,M))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Nb))) ).

% power_mult_numeral
tff(fact_690_power__mult__numeral,axiom,
    ! [A2: code_integer,M: num,Nb: num] : power_8256067586552552935nteger(power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb)) = power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(M),Nb))) ).

% power_mult_numeral
tff(fact_691_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),one) ).

% semiring_norm(69)
tff(fact_692_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit0,Nb)) ).

% semiring_norm(76)
tff(fact_693_one__le__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_694_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_695_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).

% minus_nat.diff_0
tff(fact_696_diffs0__imp__equal,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M) = zero_zero_nat )
       => ( M = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_697_diff__less__mono2,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_698_less__imp__diff__less,axiom,
    ! [J2: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_699_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ).

% Nat.diff_cancel
tff(fact_700_diff__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ).

% diff_cancel2
tff(fact_701_diff__add__inverse,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)),Nb) = M ).

% diff_add_inverse
tff(fact_702_diff__add__inverse2,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)),Nb) = M ).

% diff_add_inverse2
tff(fact_703_diff__le__mono2,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),M)) ) ).

% diff_le_mono2
tff(fact_704_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),B2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_705_diff__le__self,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),M) ).

% diff_le_self
tff(fact_706_diff__le__mono,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_707_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_708_le__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ) ).

% le_diff_iff
tff(fact_709_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K) )
        <=> ( M = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_710_diff__mult__distrib,axiom,
    ! [M: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% diff_mult_distrib
tff(fact_711_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% diff_mult_distrib2
tff(fact_712_le__num__One__iff,axiom,
    ! [Xc: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),one)
    <=> ( Xc = one ) ) ).

% le_num_One_iff
tff(fact_713_Suc__to__right,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(suc,Nb) = M )
     => ( Nb = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat)) ) ) ).

% Suc_to_right
tff(fact_714_Suc__diff__Suc,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_715_diff__less__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),aa_nat_nat(suc,M)) ).

% diff_less_Suc
tff(fact_716_diff__less,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),M) ) ) ).

% diff_less
tff(fact_717_Suc__diff__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) ) ) ).

% Suc_diff_le
tff(fact_718_diff__add__0,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = zero_zero_nat ).

% diff_add_0
tff(fact_719_less__diff__conv,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J2) ) ).

% less_diff_conv
tff(fact_720_add__diff__inverse__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_721_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C2)) ) ) ).

% diff_less_mono
tff(fact_722_less__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ) ).

% less_diff_iff
tff(fact_723_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I) = K )
      <=> ( J2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_724_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_725_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_726_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J2) ) ) ).

% Nat.le_diff_conv2
tff(fact_727_le__diff__conv,axiom,
    ! [J2: nat,K: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).

% le_diff_conv
tff(fact_728_subset__minus__empty,axiom,
    ! [A: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B)
     => ( minus_minus_set_real(A,B) = bot_bot_set_real ) ) ).

% subset_minus_empty
tff(fact_729_subset__minus__empty,axiom,
    ! [A: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),B)
     => ( minus_minus_set_o(A,B) = bot_bot_set_o ) ) ).

% subset_minus_empty
tff(fact_730_subset__minus__empty,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( minus_minus_set_int(A,B) = bot_bot_set_int ) ) ).

% subset_minus_empty
tff(fact_731_subset__minus__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( minus_minus_set_nat(A,B) = bot_bot_set_nat ) ) ).

% subset_minus_empty
tff(fact_732_diff__Suc__less,axiom,
    ! [Nb: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,I))),Nb) ) ).

% diff_Suc_less
tff(fact_733_nat__diff__split,axiom,
    ! [P: fun_nat_o,A2: nat,B2: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D3: nat] :
            ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D3) )
           => aa_nat_o(P,D3) ) ) ) ).

% nat_diff_split
tff(fact_734_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A2: nat,B2: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D3: nat] :
              ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D3) )
              & ~ aa_nat_o(P,D3) ) ) ) ).

% nat_diff_split_asm
tff(fact_735_less__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),K)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_736_div__mult2__numeral__eq,axiom,
    ! [A2: nat,K: num,L: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,K))),aa_num_nat(numeral_numeral_nat,L)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(K),L))) ).

% div_mult2_numeral_eq
tff(fact_737_div__mult2__numeral__eq,axiom,
    ! [A2: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(K),L))) ).

% div_mult2_numeral_eq
tff(fact_738_nat__eq__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),Nb) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),U)),M) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_739_nat__eq__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),Nb) )
      <=> ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_740_nat__le__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),U)),M)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_741_nat__le__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_742_nat__diff__add__eq1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),U)),M)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_743_nat__diff__add__eq2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_744_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),aa_nat_nat(suc,zero_zero_nat))),M) ) ) ).

% nz_le_conv_less
tff(fact_745_nat__less__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),U)),M)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_746_nat__less__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),M)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_747_ord__eq__le__eq__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat,D2: set_nat] :
      ( ( A2 = B2 )
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C2)
       => ( ( C2 = D2 )
         => aa_set_nat_o(ord_less_eq_set_nat(A2),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_748_ord__eq__le__eq__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( A2 = B2 )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ( C2 = D2 )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_749_ord__eq__le__eq__trans,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( A2 = B2 )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ( C2 = D2 )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_750_ord__eq__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( A2 = B2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => ( ( C2 = D2 )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_751_ord__eq__le__eq__trans,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( A2 = B2 )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => ( ( C2 = D2 )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),D2) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_752_bex2I,axiom,
    ! [A2: code_integer,B2: code_integer,S: set_Pr4811707699266497531nteger,P: fun_Co2081214437782444048eger_o] :
      ( member157494554546826820nteger(produc1086072967326762835nteger(A2,B2),S)
     => ( ( member157494554546826820nteger(produc1086072967326762835nteger(A2,B2),S)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A2),B2) )
       => ? [A3: code_integer,B3: code_integer] :
            ( member157494554546826820nteger(produc1086072967326762835nteger(A3,B3),S)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,A3),B3) ) ) ) ).

% bex2I
tff(fact_753_bex2I,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B2: set_nat,S: set_Pr3948176798113811640et_nat,P: fun_he5746958392356810273_nat_o] :
      ( member6260224972018164377et_nat(produc7507926704131184380et_nat(A2,B2),S)
     => ( ( member6260224972018164377et_nat(produc7507926704131184380et_nat(A2,B2),S)
         => aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A2),B2) )
       => ? [A3: heap_e7401611519738050253t_unit,B3: set_nat] :
            ( member6260224972018164377et_nat(produc7507926704131184380et_nat(A3,B3),S)
            & aa_set_nat_o(aa_hea8622544166608216090_nat_o(P,A3),B3) ) ) ) ).

% bex2I
tff(fact_754_bex2I,axiom,
    ! [A2: num,B2: num,S: set_Pr8218934625190621173um_num,P: fun_num_fun_num_o] :
      ( member7279096912039735102um_num(product_Pair_num_num(A2,B2),S)
     => ( ( member7279096912039735102um_num(product_Pair_num_num(A2,B2),S)
         => aa_num_o(aa_num_fun_num_o(P,A2),B2) )
       => ? [A3: num,B3: num] :
            ( member7279096912039735102um_num(product_Pair_num_num(A3,B3),S)
            & aa_num_o(aa_num_fun_num_o(P,A3),B3) ) ) ) ).

% bex2I
tff(fact_755_bex2I,axiom,
    ! [A2: nat,B2: nat,S: set_Pr1261947904930325089at_nat,P: fun_nat_fun_nat_o] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2),S)
     => ( ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A2),B2),S)
         => aa_nat_o(aa_nat_fun_nat_o(P,A2),B2) )
       => ? [A3: nat,B3: nat] :
            ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A3),B3),S)
            & aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) ) ) ) ).

% bex2I
tff(fact_756_bex2I,axiom,
    ! [A2: int,B2: int,S: set_Pr958786334691620121nt_int,P: fun_int_fun_int_o] :
      ( member5262025264175285858nt_int(product_Pair_int_int(A2,B2),S)
     => ( ( member5262025264175285858nt_int(product_Pair_int_int(A2,B2),S)
         => aa_int_o(aa_int_fun_int_o(P,A2),B2) )
       => ? [A3: int,B3: int] :
            ( member5262025264175285858nt_int(product_Pair_int_int(A3,B3),S)
            & aa_int_o(aa_int_fun_int_o(P,A3),B3) ) ) ) ).

% bex2I
tff(fact_757_Suc__inject,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(suc,Xc) = aa_nat_nat(suc,Ya) )
     => ( Xc = Ya ) ) ).

% Suc_inject
tff(fact_758_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_759_nat__neq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% nat_neq_iff
tff(fact_760_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_761_less__not__refl2,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( M != Nb ) ) ).

% less_not_refl2
tff(fact_762_less__not__refl3,axiom,
    ! [S2: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S2),Ta)
     => ( S2 != Ta ) ) ).

% less_not_refl3
tff(fact_763_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_764_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% nat_less_induct
tff(fact_765_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ~ aa_nat_o(P,N)
         => ? [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
              & ~ aa_nat_o(P,M2) ) )
     => aa_nat_o(P,Nb) ) ).

% infinite_descent
tff(fact_766_linorder__neqE__nat,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( Xc != Ya )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc) ) ) ).

% linorder_neqE_nat
tff(fact_767_set__notEmptyE,axiom,
    ! [S: set_VEBT_VEBT] :
      ( ( S != bot_bo8194388402131092736T_VEBT )
     => ~ ! [X3: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),S) ) ).

% set_notEmptyE
tff(fact_768_set__notEmptyE,axiom,
    ! [S: set_complex] :
      ( ( S != bot_bot_set_complex )
     => ~ ! [X3: complex] : ~ aa_set_complex_o(member_complex(X3),S) ) ).

% set_notEmptyE
tff(fact_769_set__notEmptyE,axiom,
    ! [S: set_real] :
      ( ( S != bot_bot_set_real )
     => ~ ! [X3: real] : ~ aa_set_real_o(member_real(X3),S) ) ).

% set_notEmptyE
tff(fact_770_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X3: $o] : ~ aa_set_o_o(member_o((X3)),S) ) ).

% set_notEmptyE
tff(fact_771_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X3: nat] : ~ aa_set_nat_o(member_nat(X3),S) ) ).

% set_notEmptyE
tff(fact_772_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X3: int] : ~ aa_set_int_o(member_int(X3),S) ) ).

% set_notEmptyE
tff(fact_773_memb__imp__not__empty,axiom,
    ! [Xc: vEBT_VEBT,S: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),S)
     => ( S != bot_bo8194388402131092736T_VEBT ) ) ).

% memb_imp_not_empty
tff(fact_774_memb__imp__not__empty,axiom,
    ! [Xc: complex,S: set_complex] :
      ( aa_set_complex_o(member_complex(Xc),S)
     => ( S != bot_bot_set_complex ) ) ).

% memb_imp_not_empty
tff(fact_775_memb__imp__not__empty,axiom,
    ! [Xc: real,S: set_real] :
      ( aa_set_real_o(member_real(Xc),S)
     => ( S != bot_bot_set_real ) ) ).

% memb_imp_not_empty
tff(fact_776_memb__imp__not__empty,axiom,
    ! [Xc: $o,S: set_o] :
      ( aa_set_o_o(member_o((Xc)),S)
     => ( S != bot_bot_set_o ) ) ).

% memb_imp_not_empty
tff(fact_777_memb__imp__not__empty,axiom,
    ! [Xc: nat,S: set_nat] :
      ( aa_set_nat_o(member_nat(Xc),S)
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
tff(fact_778_memb__imp__not__empty,axiom,
    ! [Xc: int,S: set_int] :
      ( aa_set_int_o(member_int(Xc),S)
     => ( S != bot_bot_set_int ) ) ).

% memb_imp_not_empty
tff(fact_779_size__neq__size__imp__neq,axiom,
    ! [Xc: list_VEBT_VEBT,Ya: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xc) != size_s6755466524823107622T_VEBT(Ya) )
     => ( Xc != Ya ) ) ).

% size_neq_size_imp_neq
tff(fact_780_size__neq__size__imp__neq,axiom,
    ! [Xc: list_real,Ya: list_real] :
      ( ( size_size_list_real(Xc) != size_size_list_real(Ya) )
     => ( Xc != Ya ) ) ).

% size_neq_size_imp_neq
tff(fact_781_size__neq__size__imp__neq,axiom,
    ! [Xc: list_o,Ya: list_o] :
      ( ( size_size_list_o(Xc) != size_size_list_o(Ya) )
     => ( Xc != Ya ) ) ).

% size_neq_size_imp_neq
tff(fact_782_size__neq__size__imp__neq,axiom,
    ! [Xc: list_nat,Ya: list_nat] :
      ( ( size_size_list_nat(Xc) != size_size_list_nat(Ya) )
     => ( Xc != Ya ) ) ).

% size_neq_size_imp_neq
tff(fact_783_size__neq__size__imp__neq,axiom,
    ! [Xc: num,Ya: num] :
      ( ( aa_num_nat(size_size_num,Xc) != aa_num_nat(size_size_num,Ya) )
     => ( Xc != Ya ) ) ).

% size_neq_size_imp_neq
tff(fact_784_size__neq__size__imp__neq,axiom,
    ! [Xc: list_VEBT_VEBTi,Ya: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xc) != size_s7982070591426661849_VEBTi(Ya) )
     => ( Xc != Ya ) ) ).

% size_neq_size_imp_neq
tff(fact_785_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,Xc: nat,M3: nat] :
      ( aa_nat_o(P,Xc)
     => ( ! [X3: nat] :
            ( aa_nat_o(P,X3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M3) )
       => ~ ! [M4: nat] :
              ( aa_nat_o(P,M4)
             => ~ ! [X4: nat] :
                    ( aa_nat_o(P,X4)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M4) ) ) ) ) ).

% bounded_Max_nat
tff(fact_786_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B2: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B2) )
       => ? [X3: nat] :
            ( aa_nat_o(P,X3)
            & ! [Y3: nat] :
                ( aa_nat_o(P,Y3)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),X3) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_787_nat__le__linear,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ).

% nat_le_linear
tff(fact_788_le__antisym,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( M = Nb ) ) ) ).

% le_antisym
tff(fact_789_eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% eq_imp_le
tff(fact_790_le__trans,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J2),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).

% le_trans
tff(fact_791_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_792_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_793_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_real] : size_size_list_real(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_794_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_o] : size_size_list_o(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_795_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_796_Ex__list__of__length,axiom,
    ! [Nb: nat] :
    ? [Xs2: list_VEBT_VEBTi] : size_s7982070591426661849_VEBTi(Xs2) = Nb ).

% Ex_list_of_length
tff(fact_797_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_798_neq__if__length__neq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) != size_size_list_real(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_799_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) != size_size_list_o(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_800_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_801_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xs) != size_s7982070591426661849_VEBTi(Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_802_list__update__swap,axiom,
    ! [I: nat,I4: nat,Xs: list_VEBT_VEBTi,Xc: vEBT_VEBTi,X5: vEBT_VEBTi] :
      ( ( I != I4 )
     => ( list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xc),I4,X5) = list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I4,X5),I,Xc) ) ) ).

% list_update_swap
tff(fact_803_list__update__swap,axiom,
    ! [I: nat,I4: nat,Xs: list_VEBT_VEBT,Xc: vEBT_VEBT,X5: vEBT_VEBT] :
      ( ( I != I4 )
     => ( list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xc),I4,X5) = list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I4,X5),I,Xc) ) ) ).

% list_update_swap
tff(fact_804_power__diff,axiom,
    ! [A2: code_integer,Nb: nat,M: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( power_8256067586552552935nteger(A2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = divide6298287555418463151nteger(power_8256067586552552935nteger(A2,M),power_8256067586552552935nteger(A2,Nb)) ) ) ) ).

% power_diff
tff(fact_805_power__diff,axiom,
    ! [A2: complex,Nb: nat,M: nat] :
      ( ( A2 != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_complex(power_power_complex(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A2),M)),aa_nat_complex(power_power_complex(A2),Nb)) ) ) ) ).

% power_diff
tff(fact_806_power__diff,axiom,
    ! [A2: real,Nb: nat,M: nat] :
      ( ( A2 != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ) ) ).

% power_diff
tff(fact_807_power__diff,axiom,
    ! [A2: rat,Nb: nat,M: nat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),Nb)) ) ) ) ).

% power_diff
tff(fact_808_power__diff,axiom,
    ! [A2: nat,Nb: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ) ) ) ).

% power_diff
tff(fact_809_power__diff,axiom,
    ! [A2: int,Nb: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),Nb)) ) ) ) ).

% power_diff
tff(fact_810_div__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb)) ) ).

% div_if
tff(fact_811_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_812_le__div__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_813_le__some__optE,axiom,
    ! [M: set_nat,Xc: option_set_nat] :
      ( ord_le2843612097646854710et_nat(some_set_nat(M),Xc)
     => ~ ! [M5: set_nat] :
            ( ( Xc = some_set_nat(M5) )
           => ~ aa_set_nat_o(ord_less_eq_set_nat(M),M5) ) ) ).

% le_some_optE
tff(fact_814_le__some__optE,axiom,
    ! [M: rat,Xc: option_rat] :
      ( ord_le2406147912482264968on_rat(some_rat(M),Xc)
     => ~ ! [M5: rat] :
            ( ( Xc = some_rat(M5) )
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,M),M5) ) ) ).

% le_some_optE
tff(fact_815_le__some__optE,axiom,
    ! [M: num,Xc: option_num] :
      ( ord_le6622620407824499402on_num(some_num(M),Xc)
     => ~ ! [M5: num] :
            ( ( Xc = some_num(M5) )
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),M5) ) ) ).

% le_some_optE
tff(fact_816_le__some__optE,axiom,
    ! [M: nat,Xc: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,some_nat(M)),Xc)
     => ~ ! [M5: nat] :
            ( ( Xc = some_nat(M5) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),M5) ) ) ).

% le_some_optE
tff(fact_817_le__some__optE,axiom,
    ! [M: int,Xc: option_int] :
      ( ord_le1736525451366464988on_int(some_int(M),Xc)
     => ~ ! [M5: int] :
            ( ( Xc = some_int(M5) )
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),M5) ) ) ).

% le_some_optE
tff(fact_818_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero_nat != aa_nat_nat(suc,X22) ).

% nat.distinct(1)
tff(fact_819_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_820_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_821_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa_nat_nat(suc,X22) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_822_old_Onat_Oexhaust,axiom,
    ! [Ya: nat] :
      ( ( Ya != zero_zero_nat )
     => ~ ! [Nat3: nat] : Ya != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_823_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_induct
tff(fact_824_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Nb: nat] :
      ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,X3),zero_zero_nat)
     => ( ! [Y: nat] : aa_nat_o(aa_nat_fun_nat_o(P,zero_zero_nat),aa_nat_nat(suc,Y))
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(suc,X3)),aa_nat_nat(suc,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,M),Nb) ) ) ) ).

% diff_induct
tff(fact_825_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N))
           => aa_nat_o(P,N) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_826_Suc__neq__Zero,axiom,
    ! [M: nat] : aa_nat_nat(suc,M) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_827_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_neq_Suc
tff(fact_828_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero_nat != aa_nat_nat(suc,M) ).

% Zero_not_Suc
tff(fact_829_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M4: nat] : Nb = aa_nat_nat(suc,M4) ) ).

% not0_implies_Suc
tff(fact_830_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J3)
             => ( K != aa_nat_nat(suc,J3) ) ) ) ) ).

% Nat.lessE
tff(fact_831_Suc__lessD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_lessD
tff(fact_832_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J3)
           => ( K != aa_nat_nat(suc,J3) ) ) ) ).

% Suc_lessE
tff(fact_833_Suc__lessI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( ( aa_nat_nat(suc,M) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),Nb) ) ) ).

% Suc_lessI
tff(fact_834_less__SucE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => ( M = Nb ) ) ) ).

% less_SucE
tff(fact_835_less__SucI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_836_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,Nb)
        | ? [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
            & aa_nat_o(P,I2) ) ) ) ).

% Ex_less_Suc
tff(fact_837_less__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( M = Nb ) ) ) ).

% less_Suc_eq
tff(fact_838_not__less__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M)) ) ).

% not_less_eq
tff(fact_839_Nat_OAll__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,Nb)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
           => aa_nat_o(P,I2) ) ) ) ).

% Nat.All_less_Suc
tff(fact_840_Suc__less__eq2,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),M)
    <=> ? [M6: nat] :
          ( ( M = aa_nat_nat(suc,M6) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M6) ) ) ).

% Suc_less_eq2
tff(fact_841_less__antisym,axiom,
    ! [Nb: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
       => ( M = Nb ) ) ) ).

% less_antisym
tff(fact_842_Suc__less__SucD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_less_SucD
tff(fact_843_less__trans__Suc,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_844_less__Suc__induct,axiom,
    ! [I: nat,J2: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
     => ( ! [I5: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I5),aa_nat_nat(suc,I5))
       => ( ! [I5: nat,J3: nat,K2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),J3)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I5),J3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J3),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I5),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J2) ) ) ) ).

% less_Suc_induct
tff(fact_845_strict__inc__induct,axiom,
    ! [I: nat,J2: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
     => ( ! [I5: nat] :
            ( ( J2 = aa_nat_nat(suc,I5) )
           => aa_nat_o(P,I5) )
       => ( ! [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),J2)
             => ( aa_nat_o(P,aa_nat_nat(suc,I5))
               => aa_nat_o(P,I5) ) )
         => aa_nat_o(P,I) ) ) ) ).

% strict_inc_induct
tff(fact_846_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,M: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
      <=> ( Nb = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_847_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat) ).

% bot_nat_0.extremum_strict
tff(fact_848_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr0I
tff(fact_849_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr0
tff(fact_850_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_851_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_852_gr__implies__not0,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_853_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( ~ aa_nat_o(P,N)
             => ? [M2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
                  & ~ aa_nat_o(P,M2) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_854_nat__arith_Osuc1,axiom,
    ! [A: nat,K: nat,A2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A2) )
     => ( aa_nat_nat(suc,A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_855_add__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ).

% add_Suc
tff(fact_856_add__Suc__shift,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_857_Suc__leD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% Suc_leD
tff(fact_858_le__SucE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => ( M = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_859_le__SucI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_860_Suc__le__D,axiom,
    ! [Nb: nat,M7: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M7)
     => ? [M4: nat] : M7 = aa_nat_nat(suc,M4) ) ).

% Suc_le_D
tff(fact_861_le__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        | ( M = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_862_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_863_not__less__eq__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M) ) ).

% not_less_eq_eq
tff(fact_864_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M2)),N)
             => aa_nat_o(P,M2) )
         => aa_nat_o(P,N) )
     => aa_nat_o(P,Nb) ) ).

% full_nat_induct
tff(fact_865_nat__induct__at__least,axiom,
    ! [M: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(P,M)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_866_transitive__stepwise__le,axiom,
    ! [M: nat,Nb: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X3),X3)
       => ( ! [X3: nat,Y: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X3),Y)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(R,X3),Z2) ) )
         => ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N),aa_nat_nat(suc,N))
           => aa_nat_o(aa_nat_fun_nat_o(R,M),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_867_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),Nb) = Nb ).

% plus_nat.add_0
tff(fact_868_add__eq__self__zero,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = M )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_869_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% less_eq_nat.simps(1)
tff(fact_870_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
    <=> ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
tff(fact_871_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_872_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_0_eq
tff(fact_873_add__lessD1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% add_lessD1
tff(fact_874_add__less__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ) ).

% add_less_mono
tff(fact_875_not__add__less1,axiom,
    ! [I: nat,J2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)),I) ).

% not_add_less1
tff(fact_876_not__add__less2,axiom,
    ! [J2: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),I)),I) ).

% not_add_less2
tff(fact_877_add__less__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K)) ) ).

% add_less_mono1
tff(fact_878_trans__less__add1,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),M)) ) ).

% trans_less_add1
tff(fact_879_trans__less__add2,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J2)) ) ).

% trans_less_add2
tff(fact_880_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),L) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% less_add_eq_less
tff(fact_881_exists__leI,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ( ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),Nb)
           => ~ aa_nat_o(P,N2) )
       => aa_nat_o(P,Nb) )
     => ? [N3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),Nb)
          & aa_nat_o(P,N3) ) ) ).

% exists_leI
tff(fact_882_nat__less__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & ( M != Nb ) ) ) ).

% nat_less_le
tff(fact_883_less__imp__le__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% less_imp_le_nat
tff(fact_884_le__eq__less__or__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( M = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_885_less__or__eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | ( M = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% less_or_eq_imp_le
tff(fact_886_le__neq__implies__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( ( M != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% le_neq_implies_less
tff(fact_887_less__mono__imp__le__mono,axiom,
    ! [F2: fun_nat_nat,I: nat,J2: nat] :
      ( ! [I5: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),J3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F2,I5)),aa_nat_nat(F2,J3)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F2,I)),aa_nat_nat(F2,J2)) ) ) ).

% less_mono_imp_le_mono
tff(fact_888_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys2)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P,Ys2) )
         => aa_list_VEBT_VEBT_o(P,Xs2) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_889_length__induct,axiom,
    ! [P: fun_list_real_o,Xs: list_real] :
      ( ! [Xs2: list_real] :
          ( ! [Ys2: list_real] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(Ys2)),size_size_list_real(Xs2))
             => aa_list_real_o(P,Ys2) )
         => aa_list_real_o(P,Xs2) )
     => aa_list_real_o(P,Xs) ) ).

% length_induct
tff(fact_890_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys2)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys2) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_891_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys2)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys2) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_892_length__induct,axiom,
    ! [P: fun_li984516689242532972EBTi_o,Xs: list_VEBT_VEBTi] :
      ( ! [Xs2: list_VEBT_VEBTi] :
          ( ! [Ys2: list_VEBT_VEBTi] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s7982070591426661849_VEBTi(Ys2)),size_s7982070591426661849_VEBTi(Xs2))
             => aa_list_VEBT_VEBTi_o(P,Ys2) )
         => aa_list_VEBT_VEBTi_o(P,Xs2) )
     => aa_list_VEBT_VEBTi_o(P,Xs) ) ).

% length_induct
tff(fact_893_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb) )
    <=> ( M = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_894_add__leE,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% add_leE
tff(fact_895_le__add1,axiom,
    ! [Nb: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) ).

% le_add1
tff(fact_896_le__add2,axiom,
    ! [Nb: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ).

% le_add2
tff(fact_897_add__leD1,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% add_leD1
tff(fact_898_add__leD2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% add_leD2
tff(fact_899_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N) ) ).

% le_Suc_ex
tff(fact_900_add__le__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ) ).

% add_le_mono
tff(fact_901_add__le__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K)) ) ).

% add_le_mono1
tff(fact_902_trans__le__add1,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),M)) ) ).

% trans_le_add1
tff(fact_903_trans__le__add2,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),J2)) ) ).

% trans_le_add2
tff(fact_904_nat__le__iff__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3) ) ).

% nat_le_iff_add
tff(fact_905_mult__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% mult_0
tff(fact_906_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( K = zero_zero_nat )
        | ( M = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_907_atLeastLessThan__subset__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or4029947393144176647an_rat(A2,B2)),set_or4029947393144176647an_rat(C2,D2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_908_atLeastLessThan__subset__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or1222409239386451017an_num(A2,B2)),set_or1222409239386451017an_num(C2,D2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A2)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_909_atLeastLessThan__subset__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(A2),B2)),aa_nat_set_nat(set_or4665077453230672383an_nat(C2),D2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_910_atLeastLessThan__subset__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or4662586982721622107an_int(A2,B2)),set_or4662586982721622107an_int(C2,D2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_911_atLeastLessThan__subset__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or8404916559141939852nteger(A2,B2)),set_or8404916559141939852nteger(C2,D2))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),A2)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),D2) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_912_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( set_or66887138388493659n_real(A2,B2) = set_or66887138388493659n_real(C2,D2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D2)
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_913_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( set_or4029947393144176647an_rat(A2,B2) = set_or4029947393144176647an_rat(C2,D2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2)
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_914_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( set_or1222409239386451017an_num(A2,B2) = set_or1222409239386451017an_num(C2,D2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),D2)
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_915_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A2),B2) = aa_nat_set_nat(set_or4665077453230672383an_nat(C2),D2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_916_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( set_or4662586982721622107an_int(A2,B2) = set_or4662586982721622107an_int(C2,D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_917_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( set_or8404916559141939852nteger(A2,B2) = set_or8404916559141939852nteger(C2,D2) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D2)
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_918_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( set_or66887138388493659n_real(A2,B2) = set_or66887138388493659n_real(C2,D2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D2)
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_919_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( set_or4029947393144176647an_rat(A2,B2) = set_or4029947393144176647an_rat(C2,D2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2)
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_920_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( set_or1222409239386451017an_num(A2,B2) = set_or1222409239386451017an_num(C2,D2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
       => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),D2)
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_921_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A2),B2) = aa_nat_set_nat(set_or4665077453230672383an_nat(C2),D2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_922_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( set_or4662586982721622107an_int(A2,B2) = set_or4662586982721622107an_int(C2,D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_923_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( set_or8404916559141939852nteger(A2,B2) = set_or8404916559141939852nteger(C2,D2) )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D2)
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_924_atLeastLessThan__eq__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D2)
       => ( ( set_or66887138388493659n_real(A2,B2) = set_or66887138388493659n_real(C2,D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_925_atLeastLessThan__eq__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2)
       => ( ( set_or4029947393144176647an_rat(A2,B2) = set_or4029947393144176647an_rat(C2,D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_926_atLeastLessThan__eq__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),D2)
       => ( ( set_or1222409239386451017an_num(A2,B2) = set_or1222409239386451017an_num(C2,D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_927_atLeastLessThan__eq__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
       => ( ( aa_nat_set_nat(set_or4665077453230672383an_nat(A2),B2) = aa_nat_set_nat(set_or4665077453230672383an_nat(C2),D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_928_atLeastLessThan__eq__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
       => ( ( set_or4662586982721622107an_int(A2,B2) = set_or4662586982721622107an_int(C2,D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_929_atLeastLessThan__eq__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),D2)
       => ( ( set_or8404916559141939852nteger(A2,B2) = set_or8404916559141939852nteger(C2,D2) )
        <=> ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_930_add__mult__distrib,axiom,
    ! [M: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% add_mult_distrib
tff(fact_931_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% add_mult_distrib2
tff(fact_932_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)),U)),K) ).

% left_add_mult_distrib
tff(fact_933_le__cube,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M))) ).

% le_cube
tff(fact_934_le__square,axiom,
    ! [M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),M)) ).

% le_square
tff(fact_935_mult__le__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),L)) ) ) ).

% mult_le_mono
tff(fact_936_mult__le__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),K)) ) ).

% mult_le_mono1
tff(fact_937_mult__le__mono2,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J2)) ) ).

% mult_le_mono2
tff(fact_938_list__assn__mono,axiom,
    ! [P: fun_VE905575822219544865i_assn,P2: fun_VE905575822219544865i_assn,L: list_VEBT_VEBT,L2: list_VEBT_VEBTi] :
      ( ! [X3: vEBT_VEBT,X6: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(P,X3),X6),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(P2,X3),X6))
     => entails(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P,L),L2),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(P2,L),L2)) ) ).

% list_assn_mono
tff(fact_939_lift__Suc__mono__less,axiom,
    ! [F2: fun_nat_real,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F2,N)),aa_nat_real(F2,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F2,Nb)),aa_nat_real(F2,N4)) ) ) ).

% lift_Suc_mono_less
tff(fact_940_lift__Suc__mono__less,axiom,
    ! [F2: fun_nat_rat,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F2,N)),aa_nat_rat(F2,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F2,Nb)),aa_nat_rat(F2,N4)) ) ) ).

% lift_Suc_mono_less
tff(fact_941_lift__Suc__mono__less,axiom,
    ! [F2: fun_nat_num,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F2,N)),aa_nat_num(F2,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F2,Nb)),aa_nat_num(F2,N4)) ) ) ).

% lift_Suc_mono_less
tff(fact_942_lift__Suc__mono__less,axiom,
    ! [F2: fun_nat_nat,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F2,N)),aa_nat_nat(F2,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F2,Nb)),aa_nat_nat(F2,N4)) ) ) ).

% lift_Suc_mono_less
tff(fact_943_lift__Suc__mono__less,axiom,
    ! [F2: fun_nat_int,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F2,N)),aa_nat_int(F2,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N4)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F2,Nb)),aa_nat_int(F2,N4)) ) ) ).

% lift_Suc_mono_less
tff(fact_944_lift__Suc__mono__less__iff,axiom,
    ! [F2: fun_nat_real,Nb: nat,M: nat] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F2,N)),aa_nat_real(F2,aa_nat_nat(suc,N)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F2,Nb)),aa_nat_real(F2,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_945_lift__Suc__mono__less__iff,axiom,
    ! [F2: fun_nat_rat,Nb: nat,M: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F2,N)),aa_nat_rat(F2,aa_nat_nat(suc,N)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F2,Nb)),aa_nat_rat(F2,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_946_lift__Suc__mono__less__iff,axiom,
    ! [F2: fun_nat_num,Nb: nat,M: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F2,N)),aa_nat_num(F2,aa_nat_nat(suc,N)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F2,Nb)),aa_nat_num(F2,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_947_lift__Suc__mono__less__iff,axiom,
    ! [F2: fun_nat_nat,Nb: nat,M: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F2,N)),aa_nat_nat(F2,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F2,Nb)),aa_nat_nat(F2,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_948_lift__Suc__mono__less__iff,axiom,
    ! [F2: fun_nat_int,Nb: nat,M: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F2,N)),aa_nat_int(F2,aa_nat_nat(suc,N)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F2,Nb)),aa_nat_int(F2,M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_949_lift__Suc__antimono__le,axiom,
    ! [F2: fun_nat_set_nat,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F2,aa_nat_nat(suc,N))),aa_nat_set_nat(F2,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F2,N4)),aa_nat_set_nat(F2,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_950_lift__Suc__antimono__le,axiom,
    ! [F2: fun_nat_rat,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,aa_nat_nat(suc,N))),aa_nat_rat(F2,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,N4)),aa_nat_rat(F2,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_951_lift__Suc__antimono__le,axiom,
    ! [F2: fun_nat_num,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F2,aa_nat_nat(suc,N))),aa_nat_num(F2,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F2,N4)),aa_nat_num(F2,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_952_lift__Suc__antimono__le,axiom,
    ! [F2: fun_nat_nat,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F2,aa_nat_nat(suc,N))),aa_nat_nat(F2,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F2,N4)),aa_nat_nat(F2,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_953_lift__Suc__antimono__le,axiom,
    ! [F2: fun_nat_int,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F2,aa_nat_nat(suc,N))),aa_nat_int(F2,N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F2,N4)),aa_nat_int(F2,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_954_lift__Suc__mono__le,axiom,
    ! [F2: fun_nat_set_nat,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F2,N)),aa_nat_set_nat(F2,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(F2,Nb)),aa_nat_set_nat(F2,N4)) ) ) ).

% lift_Suc_mono_le
tff(fact_955_lift__Suc__mono__le,axiom,
    ! [F2: fun_nat_rat,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,N)),aa_nat_rat(F2,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,Nb)),aa_nat_rat(F2,N4)) ) ) ).

% lift_Suc_mono_le
tff(fact_956_lift__Suc__mono__le,axiom,
    ! [F2: fun_nat_num,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F2,N)),aa_nat_num(F2,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F2,Nb)),aa_nat_num(F2,N4)) ) ) ).

% lift_Suc_mono_le
tff(fact_957_lift__Suc__mono__le,axiom,
    ! [F2: fun_nat_nat,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F2,N)),aa_nat_nat(F2,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F2,Nb)),aa_nat_nat(F2,N4)) ) ) ).

% lift_Suc_mono_le
tff(fact_958_lift__Suc__mono__le,axiom,
    ! [F2: fun_nat_int,Nb: nat,N4: nat] :
      ( ! [N: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F2,N)),aa_nat_int(F2,aa_nat_nat(suc,N)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N4)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F2,Nb)),aa_nat_int(F2,N4)) ) ) ).

% lift_Suc_mono_le
tff(fact_959_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
            & aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% Ex_less_Suc2
tff(fact_960_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M8: nat] : Nb = aa_nat_nat(suc,M8) ) ).

% gr0_conv_Suc
tff(fact_961_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% All_less_Suc2
tff(fact_962_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ? [M4: nat] : Nb = aa_nat_nat(suc,M4) ) ).

% gr0_implies_Suc
tff(fact_963_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
    <=> ( ( M = zero_zero_nat )
        | ? [J: nat] :
            ( ( M = aa_nat_nat(suc,J) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_964_add__is__1,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_965_one__is__add,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) )
    <=> ( ( ( M = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_966_nat__compl__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_compl_induct
tff(fact_967_nat__compl__induct_H,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nn),N)
               => aa_nat_o(P,Nn) )
           => aa_nat_o(P,aa_nat_nat(suc,N)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_compl_induct'
tff(fact_968_less__natE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ~ ! [Q5: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q5)) ) ).

% less_natE
tff(fact_969_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),M))) ).

% less_add_Suc1
tff(fact_970_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),I))) ).

% less_add_Suc2
tff(fact_971_less__iff__Suc__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K3)) ) ).

% less_iff_Suc_add
tff(fact_972_less__imp__Suc__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ? [K2: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K2)) ) ).

% less_imp_Suc_add
tff(fact_973_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),aa_nat_nat(suc,A2)) )
    <=> ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
tff(fact_974_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(suc,A2)) )
    <=> ( B2 = aa_nat_nat(suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_975_Suc__leI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb) ) ).

% Suc_leI
tff(fact_976_Suc__le__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_le_eq
tff(fact_977_dec__induct,axiom,
    ! [I: nat,J2: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => ( aa_nat_o(P,I)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J2)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(suc,N)) ) ) )
         => aa_nat_o(P,J2) ) ) ) ).

% dec_induct
tff(fact_978_inc__induct,axiom,
    ! [I: nat,J2: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => ( aa_nat_o(P,J2)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),J2)
               => ( aa_nat_o(P,aa_nat_nat(suc,N))
                 => aa_nat_o(P,N) ) ) )
         => aa_nat_o(P,I) ) ) ) ).

% inc_induct
tff(fact_979_Suc__le__lessD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_le_lessD
tff(fact_980_le__less__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
      <=> ( Nb = M ) ) ) ).

% le_less_Suc_eq
tff(fact_981_less__Suc__eq__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% less_Suc_eq_le
tff(fact_982_less__eq__Suc__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M) ) ).

% less_eq_Suc_le
tff(fact_983_le__imp__less__Suc,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_984_less__imp__add__positive,axiom,
    ! [I: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J2 ) ) ) ).

% less_imp_add_positive
tff(fact_985_ex__least__nat__le,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),Nb)
            & ! [I6: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I6),K2)
               => ~ aa_nat_o(P,I6) )
            & aa_nat_o(P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_986_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_987_mono__nat__linear__lb,axiom,
    ! [F2: fun_nat_nat,M: nat,K: nat] :
      ( ! [M4: nat,N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F2,M4)),aa_nat_nat(F2,N)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F2,M)),K)),aa_nat_nat(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))) ) ).

% mono_nat_linear_lb
tff(fact_988_mult__less__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J2),K)) ) ) ).

% mult_less_mono1
tff(fact_989_mult__less__mono2,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J2)) ) ) ).

% mult_less_mono2
tff(fact_990_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
      <=> ( M = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_991_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_992_mult__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) ).

% mult_Suc
tff(fact_993_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_994_mlex__bound,axiom,
    ! [A2: nat,A: nat,B2: nat,N5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),N5)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N5)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),N5)) ) ) ).

% mlex_bound
tff(fact_995_mlex__fst__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,N5: nat,B4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),N5)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B4),N5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N5)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N5)),B4)) ) ) ) ).

% mlex_fst_decrI
tff(fact_996_mlex__snd__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B4: nat,N5: nat] :
      ( ( A2 = A4 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N5)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N5)),B4)) ) ) ).

% mlex_snd_decrI
tff(fact_997_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
     => ( ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
           => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I5) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_998_nth__equalityI,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
     => ( ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
           => ( aa_nat_real(nth_real(Xs),I5) = aa_nat_real(nth_real(Ys),I5) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_999_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
     => ( ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I5)
            <=> aa_nat_o(nth_o(Ys),I5) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1000_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I5) = aa_nat_nat(nth_nat(Ys),I5) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1001_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
     => ( ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s7982070591426661849_VEBTi(Xs))
           => ( nth_VEBT_VEBTi(Xs,I5) = nth_VEBT_VEBTi(Ys,I5) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1002_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_1: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I2),X_1) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I2),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_1003_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_real_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_1: real] : aa_real_o(aa_nat_fun_real_o(P,I2),X_1) )
    <=> ? [Xs3: list_real] :
          ( ( size_size_list_real(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_real_o(aa_nat_fun_real_o(P,I2),aa_nat_real(nth_real(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_1004_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_1: $o] : aa_o_o(aa_nat_fun_o_o(P,I2),(X_1)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_o_o(aa_nat_fun_o_o(P,I2),aa_nat_o(nth_o(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_1005_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),X_1) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(nth_nat(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_1006_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na154763851189438853EBTi_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_1: vEBT_VEBTi] : aa_VEBT_VEBTi_o(aa_nat4571447880046010350EBTi_o(P,I2),X_1) )
    <=> ? [Xs3: list_VEBT_VEBTi] :
          ( ( size_s7982070591426661849_VEBTi(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_VEBT_VEBTi_o(aa_nat4571447880046010350EBTi_o(P,I2),nth_VEBT_VEBTi(Xs3,I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_1007_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( Xs = Ys )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1008_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_real,Ys: list_real] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_real(Xs) = size_size_list_real(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
           => ( aa_nat_real(nth_real(Xs),I2) = aa_nat_real(nth_real(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1009_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I2)
            <=> aa_nat_o(nth_o(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1010_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1011_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( Xs = Ys )
    <=> ( ( size_s7982070591426661849_VEBTi(Xs) = size_s7982070591426661849_VEBTi(Ys) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
           => ( nth_VEBT_VEBTi(Xs,I2) = nth_VEBT_VEBTi(Ys,I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1012_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_VE6480751511521678896_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
         => ? [Li: vEBT_VEBT] : aa_nat_o(aa_VEB2314620471664397097_nat_o(P,Li),I5) )
     => ~ ! [L3: list_VEBT_VEBT] :
            ( ( size_s6755466524823107622T_VEBT(L3) = Nb )
           => ~ ! [I6: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I6),Nb)
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L3),I6)),I6) ) ) ) ).

% obtain_list_from_elements
tff(fact_1013_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_real_fun_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
         => ? [Li: real] : aa_nat_o(aa_real_fun_nat_o(P,Li),I5) )
     => ~ ! [L3: list_real] :
            ( ( size_size_list_real(L3) = Nb )
           => ~ ! [I6: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I6),Nb)
                 => aa_nat_o(aa_real_fun_nat_o(P,aa_nat_real(nth_real(L3),I6)),I6) ) ) ) ).

% obtain_list_from_elements
tff(fact_1014_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_o_fun_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
         => ? [Li: $o] : aa_nat_o(aa_o_fun_nat_o(P,(Li)),I5) )
     => ~ ! [L3: list_o] :
            ( ( size_size_list_o(L3) = Nb )
           => ~ ! [I6: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I6),Nb)
                 => aa_nat_o(aa_o_fun_nat_o(P,aa_nat_o(nth_o(L3),I6)),I6) ) ) ) ).

% obtain_list_from_elements
tff(fact_1015_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_nat_fun_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
         => ? [Li: nat] : aa_nat_o(aa_nat_fun_nat_o(P,Li),I5) )
     => ~ ! [L3: list_nat] :
            ( ( size_size_list_nat(L3) = Nb )
           => ~ ! [I6: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I6),Nb)
                 => aa_nat_o(aa_nat_fun_nat_o(P,aa_nat_nat(nth_nat(L3),I6)),I6) ) ) ) ).

% obtain_list_from_elements
tff(fact_1016_obtain__list__from__elements,axiom,
    ! [Nb: nat,P: fun_VE6346650636051888901_nat_o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
         => ? [Li: vEBT_VEBTi] : aa_nat_o(aa_VEB8883426185042993134_nat_o(P,Li),I5) )
     => ~ ! [L3: list_VEBT_VEBTi] :
            ( ( size_s7982070591426661849_VEBTi(L3) = Nb )
           => ~ ! [I6: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I6),Nb)
                 => aa_nat_o(aa_VEB8883426185042993134_nat_o(P,nth_VEBT_VEBTi(L3,I6)),I6) ) ) ) ).

% obtain_list_from_elements
tff(fact_1017_mlex__leI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B4: nat,N5: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),A4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),B4)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),N5)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),N5)),B4)) ) ) ).

% mlex_leI
tff(fact_1018_insert__minus__eq,axiom,
    ! [Xc: vEBT_VEBT,Ya: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( Xc != Ya )
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xc,A),insert_VEBT_VEBT(Ya,bot_bo8194388402131092736T_VEBT)) = insert_VEBT_VEBT(Xc,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Ya,bot_bo8194388402131092736T_VEBT))) ) ) ).

% insert_minus_eq
tff(fact_1019_insert__minus__eq,axiom,
    ! [Xc: real,Ya: real,A: set_real] :
      ( ( Xc != Ya )
     => ( minus_minus_set_real(insert_real(Xc,A),insert_real(Ya,bot_bot_set_real)) = insert_real(Xc,minus_minus_set_real(A,insert_real(Ya,bot_bot_set_real))) ) ) ).

% insert_minus_eq
tff(fact_1020_insert__minus__eq,axiom,
    ! [Xc: $o,Ya: $o,A: set_o] :
      ( ~ ( (Xc)
        <=> (Ya) )
     => ( minus_minus_set_o(insert_o((Xc),A),insert_o((Ya),bot_bot_set_o)) = insert_o((Xc),minus_minus_set_o(A,insert_o((Ya),bot_bot_set_o))) ) ) ).

% insert_minus_eq
tff(fact_1021_insert__minus__eq,axiom,
    ! [Xc: int,Ya: int,A: set_int] :
      ( ( Xc != Ya )
     => ( minus_minus_set_int(insert_int(Xc,A),insert_int(Ya,bot_bot_set_int)) = insert_int(Xc,minus_minus_set_int(A,insert_int(Ya,bot_bot_set_int))) ) ) ).

% insert_minus_eq
tff(fact_1022_insert__minus__eq,axiom,
    ! [Xc: nat,Ya: nat,A: set_nat] :
      ( ( Xc != Ya )
     => ( minus_minus_set_nat(insert_nat(Xc,A),insert_nat(Ya,bot_bot_set_nat)) = insert_nat(Xc,minus_minus_set_nat(A,insert_nat(Ya,bot_bot_set_nat))) ) ) ).

% insert_minus_eq
tff(fact_1023_set__minus__singleton__eq,axiom,
    ! [Xc: vEBT_VEBT,X: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),X)
     => ( minus_5127226145743854075T_VEBT(X,insert_VEBT_VEBT(Xc,bot_bo8194388402131092736T_VEBT)) = X ) ) ).

% set_minus_singleton_eq
tff(fact_1024_set__minus__singleton__eq,axiom,
    ! [Xc: complex,X: set_complex] :
      ( ~ aa_set_complex_o(member_complex(Xc),X)
     => ( minus_811609699411566653omplex(X,insert_complex(Xc,bot_bot_set_complex)) = X ) ) ).

% set_minus_singleton_eq
tff(fact_1025_set__minus__singleton__eq,axiom,
    ! [Xc: real,X: set_real] :
      ( ~ aa_set_real_o(member_real(Xc),X)
     => ( minus_minus_set_real(X,insert_real(Xc,bot_bot_set_real)) = X ) ) ).

% set_minus_singleton_eq
tff(fact_1026_set__minus__singleton__eq,axiom,
    ! [Xc: $o,X: set_o] :
      ( ~ aa_set_o_o(member_o((Xc)),X)
     => ( minus_minus_set_o(X,insert_o((Xc),bot_bot_set_o)) = X ) ) ).

% set_minus_singleton_eq
tff(fact_1027_set__minus__singleton__eq,axiom,
    ! [Xc: int,X: set_int] :
      ( ~ aa_set_int_o(member_int(Xc),X)
     => ( minus_minus_set_int(X,insert_int(Xc,bot_bot_set_int)) = X ) ) ).

% set_minus_singleton_eq
tff(fact_1028_set__minus__singleton__eq,axiom,
    ! [Xc: nat,X: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xc),X)
     => ( minus_minus_set_nat(X,insert_nat(Xc,bot_bot_set_nat)) = X ) ) ).

% set_minus_singleton_eq
tff(fact_1029_remove__subset,axiom,
    ! [Xc: vEBT_VEBT,S: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),S)
     => aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(Xc,bot_bo8194388402131092736T_VEBT))),S) ) ).

% remove_subset
tff(fact_1030_remove__subset,axiom,
    ! [Xc: complex,S: set_complex] :
      ( aa_set_complex_o(member_complex(Xc),S)
     => aa_set_complex_o(ord_less_set_complex(minus_811609699411566653omplex(S,insert_complex(Xc,bot_bot_set_complex))),S) ) ).

% remove_subset
tff(fact_1031_remove__subset,axiom,
    ! [Xc: real,S: set_real] :
      ( aa_set_real_o(member_real(Xc),S)
     => aa_set_real_o(ord_less_set_real(minus_minus_set_real(S,insert_real(Xc,bot_bot_set_real))),S) ) ).

% remove_subset
tff(fact_1032_remove__subset,axiom,
    ! [Xc: $o,S: set_o] :
      ( aa_set_o_o(member_o((Xc)),S)
     => aa_set_o_o(ord_less_set_o(minus_minus_set_o(S,insert_o((Xc),bot_bot_set_o))),S) ) ).

% remove_subset
tff(fact_1033_remove__subset,axiom,
    ! [Xc: int,S: set_int] :
      ( aa_set_int_o(member_int(Xc),S)
     => aa_set_int_o(ord_less_set_int(minus_minus_set_int(S,insert_int(Xc,bot_bot_set_int))),S) ) ).

% remove_subset
tff(fact_1034_remove__subset,axiom,
    ! [Xc: nat,S: set_nat] :
      ( aa_set_nat_o(member_nat(Xc),S)
     => aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(S,insert_nat(Xc,bot_bot_set_nat))),S) ) ).

% remove_subset
tff(fact_1035_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M8: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M8),Nb)
          & aa_nat_o(P,M8) )
    <=> ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X2) ) ) ).

% ex_nat_less_eq
tff(fact_1036_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M8: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M8),Nb)
         => aa_nat_o(P,M8) )
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X2) ) ) ).

% all_nat_less_eq
tff(fact_1037_atLeastLessThan0,axiom,
    ! [M: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(M),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_1038_ex__least__nat__less,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ? [K2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Nb)
            & ! [I6: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I6),K2)
               => ~ aa_nat_o(P,I6) )
            & aa_nat_o(P,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_1039_one__less__mult,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) ) ) ).

% one_less_mult
tff(fact_1040_n__less__m__mult__n,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_1041_n__less__n__mult__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),M)) ) ) ).

% n_less_n_mult_m
tff(fact_1042_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_1043_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_1044_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xc: vEBT_VEBT,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(Xs,I,Xc)),J2) = $ite(I = J2,Xc,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J2)) ) ) ).

% nth_list_update
tff(fact_1045_nth__list__update,axiom,
    ! [I: nat,Xs: list_real,Xc: real,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_real(nth_real(list_update_real(Xs,I,Xc)),J2) = $ite(I = J2,Xc,aa_nat_real(nth_real(Xs),J2)) ) ) ).

% nth_list_update
tff(fact_1046_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,Xc: $o,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(nth_o(list_update_o(Xs,I,(Xc))),J2)
      <=> $ite(I = J2,(Xc),aa_nat_o(nth_o(Xs),J2)) ) ) ).

% nth_list_update
tff(fact_1047_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,Xc: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(list_update_nat(Xs,I,Xc)),J2) = $ite(I = J2,Xc,aa_nat_nat(nth_nat(Xs),J2)) ) ) ).

% nth_list_update
tff(fact_1048_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xc: vEBT_VEBTi,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,Xc),J2) = $ite(I = J2,Xc,nth_VEBT_VEBTi(Xs,J2)) ) ) ).

% nth_list_update
tff(fact_1049_nth__list__update_H,axiom,
    ! [L: list_VEBT_VEBT,I: nat,Xc: vEBT_VEBT,J2: nat] :
      aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(L,I,Xc)),J2) = $ite(
        ( ( I = J2 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L)) ),
        Xc,
        aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),J2) ) ).

% nth_list_update'
tff(fact_1050_nth__list__update_H,axiom,
    ! [L: list_real,I: nat,Xc: real,J2: nat] :
      aa_nat_real(nth_real(list_update_real(L,I,Xc)),J2) = $ite(
        ( ( I = J2 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L)) ),
        Xc,
        aa_nat_real(nth_real(L),J2) ) ).

% nth_list_update'
tff(fact_1051_nth__list__update_H,axiom,
    ! [L: list_o,I: nat,Xc: $o,J2: nat] :
      ( aa_nat_o(nth_o(list_update_o(L,I,(Xc))),J2)
    <=> $ite(
          ( ( I = J2 )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L)) ),
          (Xc),
          aa_nat_o(nth_o(L),J2) ) ) ).

% nth_list_update'
tff(fact_1052_nth__list__update_H,axiom,
    ! [L: list_nat,I: nat,Xc: nat,J2: nat] :
      aa_nat_nat(nth_nat(list_update_nat(L,I,Xc)),J2) = $ite(
        ( ( I = J2 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L)) ),
        Xc,
        aa_nat_nat(nth_nat(L),J2) ) ).

% nth_list_update'
tff(fact_1053_nth__list__update_H,axiom,
    ! [L: list_VEBT_VEBTi,I: nat,Xc: vEBT_VEBTi,J2: nat] :
      nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(L,I,Xc),J2) = $ite(
        ( ( I = J2 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L)) ),
        Xc,
        nth_VEBT_VEBTi(L,J2) ) ).

% nth_list_update'
tff(fact_1054_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Xc: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( ( list_u1324408373059187874T_VEBT(Xs,I,Xc) = Xs )
      <=> ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I) = Xc ) ) ) ).

% list_update_same_conv
tff(fact_1055_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_real,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( ( list_update_real(Xs,I,Xc) = Xs )
      <=> ( aa_nat_real(nth_real(Xs),I) = Xc ) ) ) ).

% list_update_same_conv
tff(fact_1056_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,Xc: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( ( list_update_o(Xs,I,(Xc)) = Xs )
      <=> ( aa_nat_o(nth_o(Xs),I)
        <=> (Xc) ) ) ) ).

% list_update_same_conv
tff(fact_1057_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( ( list_update_nat(Xs,I,Xc) = Xs )
      <=> ( aa_nat_nat(nth_nat(Xs),I) = Xc ) ) ) ).

% list_update_same_conv
tff(fact_1058_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,Xc: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( ( list_u6098035379799741383_VEBTi(Xs,I,Xc) = Xs )
      <=> ( nth_VEBT_VEBTi(Xs,I) = Xc ) ) ) ).

% list_update_same_conv
tff(fact_1059_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_1060_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn,A5: fun_VE1571388752190420990T_assn] :
      ( ( I3 = I7 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),I3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5) )
                      & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I5) )
                      & ( aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I5)) = aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A5,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L3204528365124325536T_VEBT(I3,A,Xs,Xsi) = vEBT_L3204528365124325536T_VEBT(I7,A5,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1061_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A: fun_VE6910866152645720856l_assn,A5: fun_VE6910866152645720856l_assn] :
      ( ( I3 = I7 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
         => ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),I3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5) )
                      & ( aa_nat_real(nth_real(Xsi),I5) = aa_nat_real(nth_real(Xsi2),I5) )
                      & ( aa_real_assn(aa_VEB8575627266564790985l_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5)),aa_nat_real(nth_real(Xsi),I5)) = aa_real_assn(aa_VEB8575627266564790985l_assn(A5,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5)),aa_nat_real(nth_real(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L4281036506115550016T_real(I3,A,Xs,Xsi) = vEBT_L4281036506115550016T_real(I7,A5,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1062_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_o,Xsi2: list_o,A: fun_VE2104784059010244834o_assn,A5: fun_VE2104784059010244834o_assn] :
      ( ( I3 = I7 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( size_size_list_o(Xsi) = size_size_list_o(Xsi2) )
         => ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),I3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5) )
                      & ( aa_nat_o(nth_o(Xsi),I5)
                      <=> aa_nat_o(nth_o(Xsi2),I5) )
                      & ( aa_o_assn(aa_VEB5185540838851036891o_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5)),aa_nat_o(nth_o(Xsi),I5)) = aa_o_assn(aa_VEB5185540838851036891o_assn(A5,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5)),aa_nat_o(nth_o(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L7058566406413635588VEBT_o(I3,A,Xs,Xsi) = vEBT_L7058566406413635588VEBT_o(I7,A5,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1063_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A: fun_VE4931415397359990460t_assn,A5: fun_VE4931415397359990460t_assn] :
      ( ( I3 = I7 )
     => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
       => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
         => ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),I3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
                 => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Xsi) )
                   => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5) )
                      & ( aa_nat_nat(nth_nat(Xsi),I5) = aa_nat_nat(nth_nat(Xsi2),I5) )
                      & ( aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5)),aa_nat_nat(nth_nat(Xsi),I5)) = aa_nat_assn(aa_VEB8640158415945897581t_assn(A5,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5)),aa_nat_nat(nth_nat(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L8650695023172932196BT_nat(I3,A,Xs,Xsi) = vEBT_L8650695023172932196BT_nat(I7,A5,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1064_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_real,Xs4: list_real,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: fun_re8050108611700907676T_assn,A5: fun_re8050108611700907676T_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),I3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
                 => ( ( size_size_list_real(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                   => ( ( aa_nat_real(nth_real(Xs),I5) = aa_nat_real(nth_real(Xs4),I5) )
                      & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I5) )
                      & ( aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I5)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I5)) = aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A5,aa_nat_real(nth_real(Xs4),I5)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L3095048238742455910T_VEBT(I3,A,Xs,Xsi) = vEBT_L3095048238742455910T_VEBT(I7,A5,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1065_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_real,Xs4: list_real,Xsi: list_real,Xsi2: list_real,A: fun_re6303261851654292474l_assn,A5: fun_re6303261851654292474l_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
       => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
         => ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),I3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
                 => ( ( size_size_list_real(Xs) = size_size_list_real(Xsi) )
                   => ( ( aa_nat_real(nth_real(Xs),I5) = aa_nat_real(nth_real(Xs4),I5) )
                      & ( aa_nat_real(nth_real(Xsi),I5) = aa_nat_real(nth_real(Xsi2),I5) )
                      & ( aa_real_assn(aa_rea5239198976744917947l_assn(A,aa_nat_real(nth_real(Xs),I5)),aa_nat_real(nth_real(Xsi),I5)) = aa_real_assn(aa_rea5239198976744917947l_assn(A5,aa_nat_real(nth_real(Xs4),I5)),aa_nat_real(nth_real(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L5184575500739366650l_real(I3,A,Xs,Xsi) = vEBT_L5184575500739366650l_real(I7,A5,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1066_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_real,Xs4: list_real,Xsi: list_o,Xsi2: list_o,A: fun_real_fun_o_assn,A5: fun_real_fun_o_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
       => ( ( size_size_list_o(Xsi) = size_size_list_o(Xsi2) )
         => ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),I3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
                 => ( ( size_size_list_real(Xs) = size_size_list_o(Xsi) )
                   => ( ( aa_nat_real(nth_real(Xs),I5) = aa_nat_real(nth_real(Xs4),I5) )
                      & ( aa_nat_o(nth_o(Xsi),I5)
                      <=> aa_nat_o(nth_o(Xsi2),I5) )
                      & ( aa_o_assn(aa_real_fun_o_assn(A,aa_nat_real(nth_real(Xs),I5)),aa_nat_o(nth_o(Xsi),I5)) = aa_o_assn(aa_real_fun_o_assn(A5,aa_nat_real(nth_real(Xs4),I5)),aa_nat_o(nth_o(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L7980206306069228746real_o(I3,A,Xs,Xsi) = vEBT_L7980206306069228746real_o(I7,A5,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1067_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_real,Xs4: list_real,Xsi: list_nat,Xsi2: list_nat,A: fun_re574865750389657758t_assn,A5: fun_re574865750389657758t_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
       => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
         => ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),I3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
                 => ( ( size_size_list_real(Xs) = size_size_list_nat(Xsi) )
                   => ( ( aa_nat_real(nth_real(Xs),I5) = aa_nat_real(nth_real(Xs4),I5) )
                      & ( aa_nat_nat(nth_nat(Xsi),I5) = aa_nat_nat(nth_nat(Xsi2),I5) )
                      & ( aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I5)),aa_nat_nat(nth_nat(Xsi),I5)) = aa_nat_assn(aa_real_fun_nat_assn(A5,aa_nat_real(nth_real(Xs4),I5)),aa_nat_nat(nth_nat(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L234762979517870878al_nat(I3,A,Xs,Xsi) = vEBT_L234762979517870878al_nat(I7,A5,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1068_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_real,Xs4: list_real,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi,A: fun_re560840401387337987i_assn,A5: fun_re560840401387337987i_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
       => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
         => ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),I3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
                 => ( ( size_size_list_real(Xs) = size_s7982070591426661849_VEBTi(Xsi) )
                   => ( ( aa_nat_real(nth_real(Xs),I5) = aa_nat_real(nth_real(Xs4),I5) )
                      & ( nth_VEBT_VEBTi(Xsi,I5) = nth_VEBT_VEBTi(Xsi2,I5) )
                      & ( aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I5)),nth_VEBT_VEBTi(Xsi,I5)) = aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A5,aa_nat_real(nth_real(Xs4),I5)),nth_VEBT_VEBTi(Xsi2,I5)) ) ) ) ) )
           => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I3,A,Xs),Xsi) = aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I7,A5,Xs4),Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1069_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs: list_o,Xs4: list_o,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: fun_o_4353168661238401050T_assn,A5: fun_o_4353168661238401050T_assn] :
      ( ( I3 = I7 )
     => ( ( size_size_list_o(Xs) = size_size_list_o(Xs4) )
       => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
         => ( ! [I5: nat] :
                ( aa_set_nat_o(member_nat(I5),I3)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
                 => ( ( size_size_list_o(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                   => ( ( aa_nat_o(nth_o(Xs),I5)
                      <=> aa_nat_o(nth_o(Xs4),I5) )
                      & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I5) )
                      & ( aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A,aa_nat_o(nth_o(Xs),I5)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I5)) = aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A5,aa_nat_o(nth_o(Xs4),I5)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I5)) ) ) ) ) )
           => ( vEBT_L1319876754960170684T_VEBT(I3,A,Xs,Xsi) = vEBT_L1319876754960170684T_VEBT(I7,A5,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1070_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_VE1571388752190420990T_assn,A5: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I5: nat] :
                  ( aa_set_nat_o(member_nat(I5),I3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5) )
                        & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L3204528365124325536T_VEBT(I3,A,Xs,Xsi) = vEBT_L3204528365124325536T_VEBT(I7,A5,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1071_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_VE6910866152645720856l_assn,A5: fun_VE6910866152645720856l_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
           => ( ! [I5: nat] :
                  ( aa_set_nat_o(member_nat(I5),I3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5) )
                        & ( aa_nat_real(nth_real(Xsi),I5) = aa_nat_real(nth_real(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L4281036506115550016T_real(I3,A,Xs,Xsi) = vEBT_L4281036506115550016T_real(I7,A5,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1072_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_VE2104784059010244834o_assn,A5: fun_VE2104784059010244834o_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_o,Xsi2: list_o] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( size_size_list_o(Xsi) = size_size_list_o(Xsi2) )
           => ( ! [I5: nat] :
                  ( aa_set_nat_o(member_nat(I5),I3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5) )
                        & ( aa_nat_o(nth_o(Xsi),I5)
                        <=> aa_nat_o(nth_o(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L7058566406413635588VEBT_o(I3,A,Xs,Xsi) = vEBT_L7058566406413635588VEBT_o(I7,A5,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1073_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_VE4931415397359990460t_assn,A5: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Xs4) )
         => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
           => ( ! [I5: nat] :
                  ( aa_set_nat_o(member_nat(I5),I3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
                   => ( ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Xsi) )
                     => ( ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs4),I5) )
                        & ( aa_nat_nat(nth_nat(Xsi),I5) = aa_nat_nat(nth_nat(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L8650695023172932196BT_nat(I3,A,Xs,Xsi) = vEBT_L8650695023172932196BT_nat(I7,A5,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1074_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_re8050108611700907676T_assn,A5: fun_re8050108611700907676T_assn,Xs: list_real,Xs4: list_real,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I5: nat] :
                  ( aa_set_nat_o(member_nat(I5),I3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
                   => ( ( size_size_list_real(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                     => ( ( aa_nat_real(nth_real(Xs),I5) = aa_nat_real(nth_real(Xs4),I5) )
                        & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L3095048238742455910T_VEBT(I3,A,Xs,Xsi) = vEBT_L3095048238742455910T_VEBT(I7,A5,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1075_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_re6303261851654292474l_assn,A5: fun_re6303261851654292474l_assn,Xs: list_real,Xs4: list_real,Xsi: list_real,Xsi2: list_real] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
         => ( ( size_size_list_real(Xsi) = size_size_list_real(Xsi2) )
           => ( ! [I5: nat] :
                  ( aa_set_nat_o(member_nat(I5),I3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
                   => ( ( size_size_list_real(Xs) = size_size_list_real(Xsi) )
                     => ( ( aa_nat_real(nth_real(Xs),I5) = aa_nat_real(nth_real(Xs4),I5) )
                        & ( aa_nat_real(nth_real(Xsi),I5) = aa_nat_real(nth_real(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L5184575500739366650l_real(I3,A,Xs,Xsi) = vEBT_L5184575500739366650l_real(I7,A5,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1076_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_real_fun_o_assn,A5: fun_real_fun_o_assn,Xs: list_real,Xs4: list_real,Xsi: list_o,Xsi2: list_o] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
         => ( ( size_size_list_o(Xsi) = size_size_list_o(Xsi2) )
           => ( ! [I5: nat] :
                  ( aa_set_nat_o(member_nat(I5),I3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
                   => ( ( size_size_list_real(Xs) = size_size_list_o(Xsi) )
                     => ( ( aa_nat_real(nth_real(Xs),I5) = aa_nat_real(nth_real(Xs4),I5) )
                        & ( aa_nat_o(nth_o(Xsi),I5)
                        <=> aa_nat_o(nth_o(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L7980206306069228746real_o(I3,A,Xs,Xsi) = vEBT_L7980206306069228746real_o(I7,A5,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1077_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_re574865750389657758t_assn,A5: fun_re574865750389657758t_assn,Xs: list_real,Xs4: list_real,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
         => ( ( size_size_list_nat(Xsi) = size_size_list_nat(Xsi2) )
           => ( ! [I5: nat] :
                  ( aa_set_nat_o(member_nat(I5),I3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
                   => ( ( size_size_list_real(Xs) = size_size_list_nat(Xsi) )
                     => ( ( aa_nat_real(nth_real(Xs),I5) = aa_nat_real(nth_real(Xs4),I5) )
                        & ( aa_nat_nat(nth_nat(Xsi),I5) = aa_nat_nat(nth_nat(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L234762979517870878al_nat(I3,A,Xs,Xsi) = vEBT_L234762979517870878al_nat(I7,A5,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1078_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_re560840401387337987i_assn,A5: fun_re560840401387337987i_assn,Xs: list_real,Xs4: list_real,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( size_size_list_real(Xs) = size_size_list_real(Xs4) )
         => ( ( size_s7982070591426661849_VEBTi(Xsi) = size_s7982070591426661849_VEBTi(Xsi2) )
           => ( ! [I5: nat] :
                  ( aa_set_nat_o(member_nat(I5),I3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
                   => ( ( size_size_list_real(Xs) = size_s7982070591426661849_VEBTi(Xsi) )
                     => ( ( aa_nat_real(nth_real(Xs),I5) = aa_nat_real(nth_real(Xs4),I5) )
                        & ( nth_VEBT_VEBTi(Xsi,I5) = nth_VEBT_VEBTi(Xsi2,I5) ) ) ) ) )
             => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I3,A,Xs),Xsi) = aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I7,A5,Xs4),Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1079_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: fun_o_4353168661238401050T_assn,A5: fun_o_4353168661238401050T_assn,Xs: list_o,Xs4: list_o,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( size_size_list_o(Xs) = size_size_list_o(Xs4) )
         => ( ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xsi2) )
           => ( ! [I5: nat] :
                  ( aa_set_nat_o(member_nat(I5),I3)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
                   => ( ( size_size_list_o(Xs) = size_s6755466524823107622T_VEBT(Xsi) )
                     => ( ( aa_nat_o(nth_o(Xs),I5)
                        <=> aa_nat_o(nth_o(Xs4),I5) )
                        & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I5) = aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi2),I5) ) ) ) ) )
             => ( vEBT_L1319876754960170684T_VEBT(I3,A,Xs,Xsi) = vEBT_L1319876754960170684T_VEBT(I7,A5,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1080_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L3204528365124325536T_VEBT(I3,A,list_u1324408373059187874T_VEBT(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L3204528365124325536T_VEBT(I3,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_1081_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs: list_real,A: fun_re560840401387337987i_assn,X1: real,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I3,A,list_update_real(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I3,A,Xs),Xsi) ) ) ) ).

% subst_not_in
tff(fact_1082_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs: list_real,A: fun_re8050108611700907676T_assn,X1: real,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( vEBT_L3095048238742455910T_VEBT(I3,A,list_update_real(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L3095048238742455910T_VEBT(I3,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_1083_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs: list_o,A: fun_o_9029690514724572677i_assn,X1: $o,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(I3,A,list_update_o(Xs,I,(X1))),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(I3,A,Xs),Xsi) ) ) ) ).

% subst_not_in
tff(fact_1084_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs: list_o,A: fun_o_4353168661238401050T_assn,X1: $o,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( vEBT_L1319876754960170684T_VEBT(I3,A,list_update_o(Xs,I,(X1)),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L1319876754960170684T_VEBT(I3,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_1085_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_na6467670216208870567i_assn,X1: nat,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(I3,A,list_update_nat(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(I3,A,Xs),Xsi) ) ) ) ).

% subst_not_in
tff(fact_1086_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_na4489259242115493880T_assn,X1: nat,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( vEBT_L8511957252848910786T_VEBT(I3,A,list_update_nat(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L8511957252848910786T_VEBT(I3,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_1087_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBTi,A: fun_VE2561185598974572172i_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L886525131989349516_VEBTi(I3,A,list_u6098035379799741383_VEBTi(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_lis2508312872377092907i_assn(vEBT_L886525131989349516_VEBTi(I3,A,Xs),Xsi) ) ) ) ).

% subst_not_in
tff(fact_1088_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBTi,A: fun_VE5577048379444288083T_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L2497118539674116125T_VEBT(I3,A,list_u6098035379799741383_VEBTi(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = vEBT_L2497118539674116125T_VEBT(I3,A,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_1089_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBT,A: fun_VE905575822219544865i_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I3,A,list_u1324408373059187874T_VEBT(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I3,A,Xs),Xsi) ) ) ) ).

% subst_not_in
tff(fact_1090_atLeastLessThanSuc,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb),insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_1091_listI__assn__conv,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,A: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi] :
      ( ( Nb = size_s6755466524823107622T_VEBT(Xs) )
     => ( aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb),A,Xs),Xsi) = aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A,Xs),Xsi) ) ) ).

% listI_assn_conv
tff(fact_1092_list__assn__conv__idx,axiom,
    ! [A: fun_VE905575822219544865i_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A,Xs),Xsi) = aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),A,Xs),Xsi) ).

% list_assn_conv_idx
tff(fact_1093_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L3204528365124325536T_VEBT(insert_nat(I,I3),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1094_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBT,A: fun_VE4931415397359990460t_assn,Xsi: list_nat] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L8650695023172932196BT_nat(insert_nat(I,I3),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1095_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_real,A: fun_re8050108611700907676T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( vEBT_L3095048238742455910T_VEBT(insert_nat(I,I3),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1096_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_real,A: fun_re560840401387337987i_assn,Xsi: list_VEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(insert_nat(I,I3),A,Xs),Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I3,A,Xs),Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1097_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_real,A: fun_re574865750389657758t_assn,Xsi: list_nat] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( vEBT_L234762979517870878al_nat(insert_nat(I,I3),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1098_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_o,A: fun_o_4353168661238401050T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( vEBT_L1319876754960170684T_VEBT(insert_nat(I,I3),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L1319876754960170684T_VEBT(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1099_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_o,A: fun_o_9029690514724572677i_assn,Xsi: list_VEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(insert_nat(I,I3),A,Xs),Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A,aa_nat_o(nth_o(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(I3,A,Xs),Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1100_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_o,A: fun_o_fun_nat_assn,Xsi: list_nat] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( vEBT_L2281750874075065672_o_nat(insert_nat(I,I3),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_o_fun_nat_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2281750874075065672_o_nat(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1101_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_na4489259242115493880T_assn,Xsi: list_VEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( vEBT_L8511957252848910786T_VEBT(insert_nat(I,I3),A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L8511957252848910786T_VEBT(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1102_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_na6467670216208870567i_assn,Xsi: list_VEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(insert_nat(I,I3),A,Xs),Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A,aa_nat_nat(nth_nat(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(I3,A,Xs),Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1103_listI__assn__conv_H,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,A: fun_VE905575822219544865i_assn,Xsi: list_VEBT_VEBTi,F: assn] :
      ( ( Nb = size_s6755466524823107622T_VEBT(Xs) )
     => ( aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb),A,Xs),Xsi)),F) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(A,Xs),Xsi)),F) ) ) ).

% listI_assn_conv'
tff(fact_1104_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L3204528365124325536T_VEBT(insert_nat(I,I3),A,list_u1324408373059187874T_VEBT(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,X1),X22)),vEBT_L3204528365124325536T_VEBT(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1105_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs: list_real,A: fun_re560840401387337987i_assn,X1: real,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(insert_nat(I,I3),A,list_update_real(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,X1),X22)),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I3,A,Xs),Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1106_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs: list_real,A: fun_re8050108611700907676T_assn,X1: real,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( vEBT_L3095048238742455910T_VEBT(insert_nat(I,I3),A,list_update_real(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,X1),X22)),vEBT_L3095048238742455910T_VEBT(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1107_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs: list_o,A: fun_o_9029690514724572677i_assn,X1: $o,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(insert_nat(I,I3),A,list_update_o(Xs,I,(X1))),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A,(X1)),X22)),aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(I3,A,Xs),Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1108_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs: list_o,A: fun_o_4353168661238401050T_assn,X1: $o,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( vEBT_L1319876754960170684T_VEBT(insert_nat(I,I3),A,list_update_o(Xs,I,(X1)),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A,(X1)),X22)),vEBT_L1319876754960170684T_VEBT(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1109_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_na6467670216208870567i_assn,X1: nat,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(insert_nat(I,I3),A,list_update_nat(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A,X1),X22)),aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(I3,A,Xs),Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1110_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_na4489259242115493880T_assn,X1: nat,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( vEBT_L8511957252848910786T_VEBT(insert_nat(I,I3),A,list_update_nat(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A,X1),X22)),vEBT_L8511957252848910786T_VEBT(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1111_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBTi,A: fun_VE2561185598974572172i_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L886525131989349516_VEBTi(insert_nat(I,I3),A,list_u6098035379799741383_VEBTi(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A,X1),X22)),aa_lis2508312872377092907i_assn(vEBT_L886525131989349516_VEBTi(I3,A,Xs),Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1112_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBTi,A: fun_VE5577048379444288083T_assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( vEBT_L2497118539674116125T_VEBT(insert_nat(I,I3),A,list_u6098035379799741383_VEBTi(Xs,I,X1),list_u1324408373059187874T_VEBT(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A,X1),X22)),vEBT_L2497118539674116125T_VEBT(I3,A,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1113_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBT,A: fun_VE905575822219544865i_assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(insert_nat(I,I3),A,list_u1324408373059187874T_VEBT(Xs,I,X1)),list_u6098035379799741383_VEBTi(Xsi,I,X22)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(A,X1),X22)),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I3,A,Xs),Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1114_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn,Xsi: list_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L3204528365124325536T_VEBT(I3,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1115_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_VEBT_VEBT,A: fun_VE4931415397359990460t_assn,Xsi: list_nat] :
      ( aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( vEBT_L8650695023172932196BT_nat(I3,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1116_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_real,A: fun_re8050108611700907676T_assn,Xsi: list_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( vEBT_L3095048238742455910T_VEBT(I3,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1117_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_real,A: fun_re560840401387337987i_assn,Xsi: list_VEBT_VEBTi] :
      ( aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I3,A,Xs),Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1118_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_real,A: fun_re574865750389657758t_assn,Xsi: list_nat] :
      ( aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( vEBT_L234762979517870878al_nat(I3,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1119_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_o,A: fun_o_4353168661238401050T_assn,Xsi: list_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( vEBT_L1319876754960170684T_VEBT(I3,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L1319876754960170684T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1120_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_o,A: fun_o_9029690514724572677i_assn,Xsi: list_VEBT_VEBTi] :
      ( aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(I3,A,Xs),Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A,aa_nat_o(nth_o(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1121_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_o,A: fun_o_fun_nat_assn,Xsi: list_nat] :
      ( aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( vEBT_L2281750874075065672_o_nat(I3,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_nat_assn(aa_o_fun_nat_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2281750874075065672_o_nat(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1122_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_na4489259242115493880T_assn,Xsi: list_VEBT_VEBT] :
      ( aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( vEBT_L8511957252848910786T_VEBT(I3,A,Xs,Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L8511957252848910786T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1123_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs: list_nat,A: fun_na6467670216208870567i_assn,Xsi: list_VEBT_VEBTi] :
      ( aa_set_nat_o(member_nat(I),I3)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(I3,A,Xs),Xsi) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A,aa_nat_nat(nth_nat(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1124_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I3,A,Xs,Xsi)),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1125_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L8650695023172932196BT_nat(I3,A,Xs,Xsi)),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1126_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_re8050108611700907676T_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I3,A,Xs,Xsi)),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1127_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_re560840401387337987i_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I3,A,Xs),Xsi)),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1128_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_re574865750389657758t_assn,Xs: list_real,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L234762979517870878al_nat(I3,A,Xs,Xsi)),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1129_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_o_4353168661238401050T_assn,Xs: list_o,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L1319876754960170684T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L1319876754960170684T_VEBT(I3,A,Xs,Xsi)),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1130_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_o_9029690514724572677i_assn,Xs: list_o,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A,aa_nat_o(nth_o(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(I3,A,Xs),Xsi)),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1131_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_o_fun_nat_assn,Xs: list_o,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_o_fun_nat_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2281750874075065672_o_nat(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L2281750874075065672_o_nat(I3,A,Xs,Xsi)),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1132_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_na4489259242115493880T_assn,Xs: list_nat,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L8511957252848910786T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(vEBT_L8511957252848910786T_VEBT(I3,A,Xs,Xsi)),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1133_listI__assn__reinsert,axiom,
    ! [P: assn,A: fun_na6467670216208870567i_assn,Xs: list_nat,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,Q: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A,aa_nat_nat(nth_nat(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( entails(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(I3,A,Xs),Xsi)),F),Q)
           => entails(P,Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1134_arith__geo__mean,axiom,
    ! [U: real,Xc: real,Ya: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_1135_arith__geo__mean,axiom,
    ! [U: rat,Xc: rat,Ya: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(times_times_rat(Xc),Ya) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xc),Ya)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_1136_nat__div__eq__Suc__0__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),M) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_1137_ent__pure__pre__iff,axiom,
    ! [P: assn,B2: $o,Q: assn] :
      ( entails(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),Q)
    <=> ( (B2)
       => entails(P,Q) ) ) ).

% ent_pure_pre_iff
tff(fact_1138_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xc: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1139_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xc: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_1140_sum__squares__bound,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xc)),Ya)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_1141_sum__squares__bound,axiom,
    ! [Xc: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xc)),Ya)),aa_rat_rat(plus_plus_rat(aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_1142_insert__Diff__single,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] : insert_VEBT_VEBT(A2,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = insert_VEBT_VEBT(A2,A) ).

% insert_Diff_single
tff(fact_1143_insert__Diff__single,axiom,
    ! [A2: real,A: set_real] : insert_real(A2,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = insert_real(A2,A) ).

% insert_Diff_single
tff(fact_1144_insert__Diff__single,axiom,
    ! [A2: $o,A: set_o] : insert_o((A2),minus_minus_set_o(A,insert_o((A2),bot_bot_set_o))) = insert_o((A2),A) ).

% insert_Diff_single
tff(fact_1145_insert__Diff__single,axiom,
    ! [A2: int,A: set_int] : insert_int(A2,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = insert_int(A2,A) ).

% insert_Diff_single
tff(fact_1146_insert__Diff__single,axiom,
    ! [A2: nat,A: set_nat] : insert_nat(A2,minus_minus_set_nat(A,insert_nat(A2,bot_bot_set_nat))) = insert_nat(A2,A) ).

% insert_Diff_single
tff(fact_1147_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A2)),aa_complex_complex(times_times_complex(C2),B2)) = $ite(C2 = zero_zero_complex,zero_zero_complex,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1148_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: real,A2: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) = $ite(C2 = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1149_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)) = $ite(C2 = zero_zero_rat,zero_zero_rat,aa_rat_rat(divide_divide_rat(A2),B2)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1150_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A2)),aa_complex_complex(times_times_complex(C2),B2)) = aa_complex_complex(divide1717551699836669952omplex(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1151_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1152_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)) = aa_rat_rat(divide_divide_rat(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1153_nonzero__mult__div__cancel__left,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1154_nonzero__mult__div__cancel__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1155_nonzero__mult__div__cancel__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1156_nonzero__mult__div__cancel__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1157_nonzero__mult__div__cancel__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A2),B2)),A2) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1158_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C2),A2)),aa_complex_complex(times_times_complex(B2),C2)) = aa_complex_complex(divide1717551699836669952omplex(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1159_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1160_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(B2),C2)) = aa_rat_rat(divide_divide_rat(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1161_empty__Collect__eq,axiom,
    ! [P: fun_complex_o] :
      ( ( bot_bot_set_complex = collect_complex(P) )
    <=> ! [X2: complex] : ~ aa_complex_o(P,X2) ) ).

% empty_Collect_eq
tff(fact_1162_empty__Collect__eq,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] :
      ( ( bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(P) )
    <=> ! [X2: product_prod_int_int] : ~ aa_Pro4592068669816069595_int_o(P,X2) ) ).

% empty_Collect_eq
tff(fact_1163_empty__Collect__eq,axiom,
    ! [P: fun_real_o] :
      ( ( bot_bot_set_real = collect_real(P) )
    <=> ! [X2: real] : ~ aa_real_o(P,X2) ) ).

% empty_Collect_eq
tff(fact_1164_empty__Collect__eq,axiom,
    ! [P: fun_o_o] :
      ( ( bot_bot_set_o = collect_o(P) )
    <=> ! [X2: $o] : ~ aa_o_o(P,(X2)) ) ).

% empty_Collect_eq
tff(fact_1165_empty__Collect__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( bot_bot_set_nat = collect_nat(P) )
    <=> ! [X2: nat] : ~ aa_nat_o(P,X2) ) ).

% empty_Collect_eq
tff(fact_1166_empty__Collect__eq,axiom,
    ! [P: fun_int_o] :
      ( ( bot_bot_set_int = collect_int(P) )
    <=> ! [X2: int] : ~ aa_int_o(P,X2) ) ).

% empty_Collect_eq
tff(fact_1167_Collect__empty__eq,axiom,
    ! [P: fun_complex_o] :
      ( ( collect_complex(P) = bot_bot_set_complex )
    <=> ! [X2: complex] : ~ aa_complex_o(P,X2) ) ).

% Collect_empty_eq
tff(fact_1168_Collect__empty__eq,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] :
      ( ( collec213857154873943460nt_int(P) = bot_bo1796632182523588997nt_int )
    <=> ! [X2: product_prod_int_int] : ~ aa_Pro4592068669816069595_int_o(P,X2) ) ).

% Collect_empty_eq
tff(fact_1169_Collect__empty__eq,axiom,
    ! [P: fun_real_o] :
      ( ( collect_real(P) = bot_bot_set_real )
    <=> ! [X2: real] : ~ aa_real_o(P,X2) ) ).

% Collect_empty_eq
tff(fact_1170_Collect__empty__eq,axiom,
    ! [P: fun_o_o] :
      ( ( collect_o(P) = bot_bot_set_o )
    <=> ! [X2: $o] : ~ aa_o_o(P,(X2)) ) ).

% Collect_empty_eq
tff(fact_1171_Collect__empty__eq,axiom,
    ! [P: fun_nat_o] :
      ( ( collect_nat(P) = bot_bot_set_nat )
    <=> ! [X2: nat] : ~ aa_nat_o(P,X2) ) ).

% Collect_empty_eq
tff(fact_1172_Collect__empty__eq,axiom,
    ! [P: fun_int_o] :
      ( ( collect_int(P) = bot_bot_set_int )
    <=> ! [X2: int] : ~ aa_int_o(P,X2) ) ).

% Collect_empty_eq
tff(fact_1173_all__not__in__conv,axiom,
    ! [A: set_VEBT_VEBT] :
      ( ! [X2: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A)
    <=> ( A = bot_bo8194388402131092736T_VEBT ) ) ).

% all_not_in_conv
tff(fact_1174_all__not__in__conv,axiom,
    ! [A: set_complex] :
      ( ! [X2: complex] : ~ aa_set_complex_o(member_complex(X2),A)
    <=> ( A = bot_bot_set_complex ) ) ).

% all_not_in_conv
tff(fact_1175_all__not__in__conv,axiom,
    ! [A: set_real] :
      ( ! [X2: real] : ~ aa_set_real_o(member_real(X2),A)
    <=> ( A = bot_bot_set_real ) ) ).

% all_not_in_conv
tff(fact_1176_all__not__in__conv,axiom,
    ! [A: set_o] :
      ( ! [X2: $o] : ~ aa_set_o_o(member_o((X2)),A)
    <=> ( A = bot_bot_set_o ) ) ).

% all_not_in_conv
tff(fact_1177_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ! [X2: nat] : ~ aa_set_nat_o(member_nat(X2),A)
    <=> ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
tff(fact_1178_all__not__in__conv,axiom,
    ! [A: set_int] :
      ( ! [X2: int] : ~ aa_set_int_o(member_int(X2),A)
    <=> ( A = bot_bot_set_int ) ) ).

% all_not_in_conv
tff(fact_1179_empty__iff,axiom,
    ! [C2: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),bot_bo8194388402131092736T_VEBT) ).

% empty_iff
tff(fact_1180_empty__iff,axiom,
    ! [C2: complex] : ~ aa_set_complex_o(member_complex(C2),bot_bot_set_complex) ).

% empty_iff
tff(fact_1181_empty__iff,axiom,
    ! [C2: real] : ~ aa_set_real_o(member_real(C2),bot_bot_set_real) ).

% empty_iff
tff(fact_1182_empty__iff,axiom,
    ! [C2: $o] : ~ aa_set_o_o(member_o((C2)),bot_bot_set_o) ).

% empty_iff
tff(fact_1183_empty__iff,axiom,
    ! [C2: nat] : ~ aa_set_nat_o(member_nat(C2),bot_bot_set_nat) ).

% empty_iff
tff(fact_1184_empty__iff,axiom,
    ! [C2: int] : ~ aa_set_int_o(member_int(C2),bot_bot_set_int) ).

% empty_iff
tff(fact_1185_subsetI,axiom,
    ! [A: set_real,B: set_real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),A)
         => aa_set_real_o(member_real(X3),B) )
     => aa_set_real_o(ord_less_eq_set_real(A),B) ) ).

% subsetI
tff(fact_1186_subsetI,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),A)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),B) )
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B) ) ).

% subsetI
tff(fact_1187_subsetI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),A)
         => aa_set_int_o(member_int(X3),B) )
     => aa_set_int_o(ord_less_eq_set_int(A),B) ) ).

% subsetI
tff(fact_1188_subsetI,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),A)
         => aa_set_complex_o(member_complex(X3),B) )
     => aa_set_complex_o(ord_le211207098394363844omplex(A),B) ) ).

% subsetI
tff(fact_1189_subsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),A)
         => aa_set_nat_o(member_nat(X3),B) )
     => aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% subsetI
tff(fact_1190_subset__antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),A)
       => ( A = B ) ) ) ).

% subset_antisym
tff(fact_1191_insert__absorb2,axiom,
    ! [Xc: nat,A: set_nat] : insert_nat(Xc,insert_nat(Xc,A)) = insert_nat(Xc,A) ).

% insert_absorb2
tff(fact_1192_insert__absorb2,axiom,
    ! [Xc: int,A: set_int] : insert_int(Xc,insert_int(Xc,A)) = insert_int(Xc,A) ).

% insert_absorb2
tff(fact_1193_insert__absorb2,axiom,
    ! [Xc: vEBT_VEBT,A: set_VEBT_VEBT] : insert_VEBT_VEBT(Xc,insert_VEBT_VEBT(Xc,A)) = insert_VEBT_VEBT(Xc,A) ).

% insert_absorb2
tff(fact_1194_insert__absorb2,axiom,
    ! [Xc: real,A: set_real] : insert_real(Xc,insert_real(Xc,A)) = insert_real(Xc,A) ).

% insert_absorb2
tff(fact_1195_insert__absorb2,axiom,
    ! [Xc: $o,A: set_o] : insert_o((Xc),insert_o((Xc),A)) = insert_o((Xc),A) ).

% insert_absorb2
tff(fact_1196_insert__iff,axiom,
    ! [A2: $o,B2: $o,A: set_o] :
      ( aa_set_o_o(member_o((A2)),insert_o((B2),A))
    <=> ( ( (A2)
        <=> (B2) )
        | aa_set_o_o(member_o((A2)),A) ) ) ).

% insert_iff
tff(fact_1197_insert__iff,axiom,
    ! [A2: nat,B2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),insert_nat(B2,A))
    <=> ( ( A2 = B2 )
        | aa_set_nat_o(member_nat(A2),A) ) ) ).

% insert_iff
tff(fact_1198_insert__iff,axiom,
    ! [A2: real,B2: real,A: set_real] :
      ( aa_set_real_o(member_real(A2),insert_real(B2,A))
    <=> ( ( A2 = B2 )
        | aa_set_real_o(member_real(A2),A) ) ) ).

% insert_iff
tff(fact_1199_insert__iff,axiom,
    ! [A2: vEBT_VEBT,B2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),insert_VEBT_VEBT(B2,A))
    <=> ( ( A2 = B2 )
        | aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),A) ) ) ).

% insert_iff
tff(fact_1200_insert__iff,axiom,
    ! [A2: int,B2: int,A: set_int] :
      ( aa_set_int_o(member_int(A2),insert_int(B2,A))
    <=> ( ( A2 = B2 )
        | aa_set_int_o(member_int(A2),A) ) ) ).

% insert_iff
tff(fact_1201_insert__iff,axiom,
    ! [A2: complex,B2: complex,A: set_complex] :
      ( aa_set_complex_o(member_complex(A2),insert_complex(B2,A))
    <=> ( ( A2 = B2 )
        | aa_set_complex_o(member_complex(A2),A) ) ) ).

% insert_iff
tff(fact_1202_insertCI,axiom,
    ! [A2: $o,B: set_o,B2: $o] :
      ( ( ~ aa_set_o_o(member_o((A2)),B)
       => ( (A2)
        <=> (B2) ) )
     => aa_set_o_o(member_o((A2)),insert_o((B2),B)) ) ).

% insertCI
tff(fact_1203_insertCI,axiom,
    ! [A2: nat,B: set_nat,B2: nat] :
      ( ( ~ aa_set_nat_o(member_nat(A2),B)
       => ( A2 = B2 ) )
     => aa_set_nat_o(member_nat(A2),insert_nat(B2,B)) ) ).

% insertCI
tff(fact_1204_insertCI,axiom,
    ! [A2: real,B: set_real,B2: real] :
      ( ( ~ aa_set_real_o(member_real(A2),B)
       => ( A2 = B2 ) )
     => aa_set_real_o(member_real(A2),insert_real(B2,B)) ) ).

% insertCI
tff(fact_1205_insertCI,axiom,
    ! [A2: vEBT_VEBT,B: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),B)
       => ( A2 = B2 ) )
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),insert_VEBT_VEBT(B2,B)) ) ).

% insertCI
tff(fact_1206_insertCI,axiom,
    ! [A2: int,B: set_int,B2: int] :
      ( ( ~ aa_set_int_o(member_int(A2),B)
       => ( A2 = B2 ) )
     => aa_set_int_o(member_int(A2),insert_int(B2,B)) ) ).

% insertCI
tff(fact_1207_insertCI,axiom,
    ! [A2: complex,B: set_complex,B2: complex] :
      ( ( ~ aa_set_complex_o(member_complex(A2),B)
       => ( A2 = B2 ) )
     => aa_set_complex_o(member_complex(A2),insert_complex(B2,B)) ) ).

% insertCI
tff(fact_1208_Diff__idemp,axiom,
    ! [A: set_nat,B: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A,B),B) = minus_minus_set_nat(A,B) ).

% Diff_idemp
tff(fact_1209_Diff__iff,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A,B))
    <=> ( aa_set_real_o(member_real(C2),A)
        & ~ aa_set_real_o(member_real(C2),B) ) ) ).

% Diff_iff
tff(fact_1210_Diff__iff,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A,B))
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A)
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B) ) ) ).

% Diff_iff
tff(fact_1211_Diff__iff,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),minus_minus_set_int(A,B))
    <=> ( aa_set_int_o(member_int(C2),A)
        & ~ aa_set_int_o(member_int(C2),B) ) ) ).

% Diff_iff
tff(fact_1212_Diff__iff,axiom,
    ! [C2: complex,A: set_complex,B: set_complex] :
      ( aa_set_complex_o(member_complex(C2),minus_811609699411566653omplex(A,B))
    <=> ( aa_set_complex_o(member_complex(C2),A)
        & ~ aa_set_complex_o(member_complex(C2),B) ) ) ).

% Diff_iff
tff(fact_1213_Diff__iff,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),minus_minus_set_nat(A,B))
    <=> ( aa_set_nat_o(member_nat(C2),A)
        & ~ aa_set_nat_o(member_nat(C2),B) ) ) ).

% Diff_iff
tff(fact_1214_DiffI,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),A)
     => ( ~ aa_set_real_o(member_real(C2),B)
       => aa_set_real_o(member_real(C2),minus_minus_set_real(A,B)) ) ) ).

% DiffI
tff(fact_1215_DiffI,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A,B)) ) ) ).

% DiffI
tff(fact_1216_DiffI,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),A)
     => ( ~ aa_set_int_o(member_int(C2),B)
       => aa_set_int_o(member_int(C2),minus_minus_set_int(A,B)) ) ) ).

% DiffI
tff(fact_1217_DiffI,axiom,
    ! [C2: complex,A: set_complex,B: set_complex] :
      ( aa_set_complex_o(member_complex(C2),A)
     => ( ~ aa_set_complex_o(member_complex(C2),B)
       => aa_set_complex_o(member_complex(C2),minus_811609699411566653omplex(A,B)) ) ) ).

% DiffI
tff(fact_1218_DiffI,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),A)
     => ( ~ aa_set_nat_o(member_nat(C2),B)
       => aa_set_nat_o(member_nat(C2),minus_minus_set_nat(A,B)) ) ) ).

% DiffI
tff(fact_1219_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( aa_o_assn(pure_assn,(P)) = aa_o_assn(pure_assn,(Q)) )
    <=> ( (P)
      <=> (Q) ) ) ).

% pure_assn_eq_conv
tff(fact_1220_mult__cancel__right,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A2),C2) = aa_complex_complex(times_times_complex(B2),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1221_mult__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1222_mult__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),C2) = aa_rat_rat(times_times_rat(B2),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1223_mult__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1224_mult__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( aa_int_int(times_times_int(A2),C2) = aa_int_int(times_times_int(B2),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1225_mult__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(C2),A2) = aa_complex_complex(times_times_complex(C2),B2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1226_mult__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1227_mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(C2),A2) = aa_rat_rat(times_times_rat(C2),B2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1228_mult__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2) )
    <=> ( ( C2 = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1229_mult__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( aa_int_int(times_times_int(C2),A2) = aa_int_int(times_times_int(C2),B2) )
    <=> ( ( C2 = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1230_mult__eq__0__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A2),B2) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
tff(fact_1231_mult__eq__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_1232_mult__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),B2) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1233_mult__eq__0__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_1234_mult__eq__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(times_times_int(A2),B2) = zero_zero_int )
    <=> ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_1235_mult__zero__right,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(A2),zero_zero_complex) = zero_zero_complex ).

% mult_zero_right
tff(fact_1236_mult__zero__right,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),zero_zero_real) = zero_zero_real ).

% mult_zero_right
tff(fact_1237_mult__zero__right,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1238_mult__zero__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_1239_mult__zero__right,axiom,
    ! [A2: int] : aa_int_int(times_times_int(A2),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_1240_mult__zero__left,axiom,
    ! [A2: complex] : aa_complex_complex(times_times_complex(zero_zero_complex),A2) = zero_zero_complex ).

% mult_zero_left
tff(fact_1241_mult__zero__left,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_zero_real),A2) = zero_zero_real ).

% mult_zero_left
tff(fact_1242_mult__zero__left,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(zero_zero_rat),A2) = zero_zero_rat ).

% mult_zero_left
tff(fact_1243_mult__zero__left,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A2) = zero_zero_nat ).

% mult_zero_left
tff(fact_1244_mult__zero__left,axiom,
    ! [A2: int] : aa_int_int(times_times_int(zero_zero_int),A2) = zero_zero_int ).

% mult_zero_left
tff(fact_1245_division__ring__divide__zero,axiom,
    ! [A2: complex] : aa_complex_complex(divide1717551699836669952omplex(A2),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_1246_division__ring__divide__zero,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),zero_zero_real) = zero_zero_real ).

% division_ring_divide_zero
tff(fact_1247_division__ring__divide__zero,axiom,
    ! [A2: rat] : aa_rat_rat(divide_divide_rat(A2),zero_zero_rat) = zero_zero_rat ).

% division_ring_divide_zero
tff(fact_1248_divide__cancel__right,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A2),C2) = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
tff(fact_1249_divide__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
tff(fact_1250_divide__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A2),C2) = aa_rat_rat(divide_divide_rat(B2),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
tff(fact_1251_divide__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C2),A2) = aa_complex_complex(divide1717551699836669952omplex(C2),B2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
tff(fact_1252_divide__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2) )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
tff(fact_1253_divide__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C2),A2) = aa_rat_rat(divide_divide_rat(C2),B2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
tff(fact_1254_div__by__0,axiom,
    ! [A2: complex] : aa_complex_complex(divide1717551699836669952omplex(A2),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1255_div__by__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),zero_zero_real) = zero_zero_real ).

% div_by_0
tff(fact_1256_div__by__0,axiom,
    ! [A2: rat] : aa_rat_rat(divide_divide_rat(A2),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1257_div__by__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),zero_zero_nat) = zero_zero_nat ).

% div_by_0
tff(fact_1258_div__by__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1259_divide__eq__0__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A2),B2) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_1260_divide__eq__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) = zero_zero_real )
    <=> ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_1261_divide__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A2),B2) = zero_zero_rat )
    <=> ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_1262_div__0,axiom,
    ! [A2: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A2) = zero_zero_complex ).

% div_0
tff(fact_1263_div__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,zero_zero_real),A2) = zero_zero_real ).

% div_0
tff(fact_1264_div__0,axiom,
    ! [A2: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A2) = zero_zero_rat ).

% div_0
tff(fact_1265_div__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A2) = zero_zero_nat ).

% div_0
tff(fact_1266_div__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A2) = zero_zero_int ).

% div_0
tff(fact_1267_times__divide__eq__right,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(times_times_complex(A2),aa_complex_complex(divide1717551699836669952omplex(B2),C2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A2),B2)),C2) ).

% times_divide_eq_right
tff(fact_1268_times__divide__eq__right,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),C2) ).

% times_divide_eq_right
tff(fact_1269_times__divide__eq__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(divide_divide_rat(B2),C2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),B2)),C2) ).

% times_divide_eq_right
tff(fact_1270_divide__divide__eq__right,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(A2),aa_complex_complex(divide1717551699836669952omplex(B2),C2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A2),C2)),B2) ).

% divide_divide_eq_right
tff(fact_1271_divide__divide__eq__right,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2) ).

% divide_divide_eq_right
tff(fact_1272_divide__divide__eq__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(divide_divide_rat(B2),C2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),C2)),B2) ).

% divide_divide_eq_right
tff(fact_1273_divide__divide__eq__left,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A2),B2)),C2) = aa_complex_complex(divide1717551699836669952omplex(A2),aa_complex_complex(times_times_complex(B2),C2)) ).

% divide_divide_eq_left
tff(fact_1274_divide__divide__eq__left,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% divide_divide_eq_left
tff(fact_1275_divide__divide__eq__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A2),B2)),C2) = aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(times_times_rat(B2),C2)) ).

% divide_divide_eq_left
tff(fact_1276_times__divide__eq__left,axiom,
    ! [B2: complex,C2: complex,A2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B2),C2)),A2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B2),A2)),C2) ).

% times_divide_eq_left
tff(fact_1277_times__divide__eq__left,axiom,
    ! [B2: real,C2: real,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2)),C2) ).

% times_divide_eq_left
tff(fact_1278_times__divide__eq__left,axiom,
    ! [B2: rat,C2: rat,A2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(B2),C2)),A2) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(B2),A2)),C2) ).

% times_divide_eq_left
tff(fact_1279_empty__subsetI,axiom,
    ! [A: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A) ).

% empty_subsetI
tff(fact_1280_empty__subsetI,axiom,
    ! [A: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A) ).

% empty_subsetI
tff(fact_1281_empty__subsetI,axiom,
    ! [A: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A) ).

% empty_subsetI
tff(fact_1282_empty__subsetI,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A) ).

% empty_subsetI
tff(fact_1283_subset__empty,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),bot_bot_set_real)
    <=> ( A = bot_bot_set_real ) ) ).

% subset_empty
tff(fact_1284_subset__empty,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),bot_bot_set_o)
    <=> ( A = bot_bot_set_o ) ) ).

% subset_empty
tff(fact_1285_subset__empty,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% subset_empty
tff(fact_1286_subset__empty,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% subset_empty
tff(fact_1287_singletonI,axiom,
    ! [A2: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)) ).

% singletonI
tff(fact_1288_singletonI,axiom,
    ! [A2: complex] : aa_set_complex_o(member_complex(A2),insert_complex(A2,bot_bot_set_complex)) ).

% singletonI
tff(fact_1289_singletonI,axiom,
    ! [A2: real] : aa_set_real_o(member_real(A2),insert_real(A2,bot_bot_set_real)) ).

% singletonI
tff(fact_1290_singletonI,axiom,
    ! [A2: $o] : aa_set_o_o(member_o((A2)),insert_o((A2),bot_bot_set_o)) ).

% singletonI
tff(fact_1291_singletonI,axiom,
    ! [A2: nat] : aa_set_nat_o(member_nat(A2),insert_nat(A2,bot_bot_set_nat)) ).

% singletonI
tff(fact_1292_singletonI,axiom,
    ! [A2: int] : aa_set_int_o(member_int(A2),insert_int(A2,bot_bot_set_int)) ).

% singletonI
tff(fact_1293_insert__subset,axiom,
    ! [Xc: $o,A: set_o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(insert_o((Xc),A)),B)
    <=> ( aa_set_o_o(member_o((Xc)),B)
        & aa_set_o_o(ord_less_eq_set_o(A),B) ) ) ).

% insert_subset
tff(fact_1294_insert__subset,axiom,
    ! [Xc: real,A: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(insert_real(Xc,A)),B)
    <=> ( aa_set_real_o(member_real(Xc),B)
        & aa_set_real_o(ord_less_eq_set_real(A),B) ) ) ).

% insert_subset
tff(fact_1295_insert__subset,axiom,
    ! [Xc: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(Xc,A)),B)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),B)
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B) ) ) ).

% insert_subset
tff(fact_1296_insert__subset,axiom,
    ! [Xc: int,A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(insert_int(Xc,A)),B)
    <=> ( aa_set_int_o(member_int(Xc),B)
        & aa_set_int_o(ord_less_eq_set_int(A),B) ) ) ).

% insert_subset
tff(fact_1297_insert__subset,axiom,
    ! [Xc: complex,A: set_complex,B: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(insert_complex(Xc,A)),B)
    <=> ( aa_set_complex_o(member_complex(Xc),B)
        & aa_set_complex_o(ord_le211207098394363844omplex(A),B) ) ) ).

% insert_subset
tff(fact_1298_insert__subset,axiom,
    ! [Xc: nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(insert_nat(Xc,A)),B)
    <=> ( aa_set_nat_o(member_nat(Xc),B)
        & aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ) ).

% insert_subset
tff(fact_1299_Diff__cancel,axiom,
    ! [A: set_real] : minus_minus_set_real(A,A) = bot_bot_set_real ).

% Diff_cancel
tff(fact_1300_Diff__cancel,axiom,
    ! [A: set_o] : minus_minus_set_o(A,A) = bot_bot_set_o ).

% Diff_cancel
tff(fact_1301_Diff__cancel,axiom,
    ! [A: set_int] : minus_minus_set_int(A,A) = bot_bot_set_int ).

% Diff_cancel
tff(fact_1302_Diff__cancel,axiom,
    ! [A: set_nat] : minus_minus_set_nat(A,A) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_1303_empty__Diff,axiom,
    ! [A: set_real] : minus_minus_set_real(bot_bot_set_real,A) = bot_bot_set_real ).

% empty_Diff
tff(fact_1304_empty__Diff,axiom,
    ! [A: set_o] : minus_minus_set_o(bot_bot_set_o,A) = bot_bot_set_o ).

% empty_Diff
tff(fact_1305_empty__Diff,axiom,
    ! [A: set_int] : minus_minus_set_int(bot_bot_set_int,A) = bot_bot_set_int ).

% empty_Diff
tff(fact_1306_empty__Diff,axiom,
    ! [A: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A) = bot_bot_set_nat ).

% empty_Diff
tff(fact_1307_Diff__empty,axiom,
    ! [A: set_real] : minus_minus_set_real(A,bot_bot_set_real) = A ).

% Diff_empty
tff(fact_1308_Diff__empty,axiom,
    ! [A: set_o] : minus_minus_set_o(A,bot_bot_set_o) = A ).

% Diff_empty
tff(fact_1309_Diff__empty,axiom,
    ! [A: set_int] : minus_minus_set_int(A,bot_bot_set_int) = A ).

% Diff_empty
tff(fact_1310_Diff__empty,axiom,
    ! [A: set_nat] : minus_minus_set_nat(A,bot_bot_set_nat) = A ).

% Diff_empty
tff(fact_1311_insert__Diff1,axiom,
    ! [Xc: $o,B: set_o,A: set_o] :
      ( aa_set_o_o(member_o((Xc)),B)
     => ( minus_minus_set_o(insert_o((Xc),A),B) = minus_minus_set_o(A,B) ) ) ).

% insert_Diff1
tff(fact_1312_insert__Diff1,axiom,
    ! [Xc: real,B: set_real,A: set_real] :
      ( aa_set_real_o(member_real(Xc),B)
     => ( minus_minus_set_real(insert_real(Xc,A),B) = minus_minus_set_real(A,B) ) ) ).

% insert_Diff1
tff(fact_1313_insert__Diff1,axiom,
    ! [Xc: vEBT_VEBT,B: set_VEBT_VEBT,A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),B)
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xc,A),B) = minus_5127226145743854075T_VEBT(A,B) ) ) ).

% insert_Diff1
tff(fact_1314_insert__Diff1,axiom,
    ! [Xc: int,B: set_int,A: set_int] :
      ( aa_set_int_o(member_int(Xc),B)
     => ( minus_minus_set_int(insert_int(Xc,A),B) = minus_minus_set_int(A,B) ) ) ).

% insert_Diff1
tff(fact_1315_insert__Diff1,axiom,
    ! [Xc: complex,B: set_complex,A: set_complex] :
      ( aa_set_complex_o(member_complex(Xc),B)
     => ( minus_811609699411566653omplex(insert_complex(Xc,A),B) = minus_811609699411566653omplex(A,B) ) ) ).

% insert_Diff1
tff(fact_1316_insert__Diff1,axiom,
    ! [Xc: nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(Xc),B)
     => ( minus_minus_set_nat(insert_nat(Xc,A),B) = minus_minus_set_nat(A,B) ) ) ).

% insert_Diff1
tff(fact_1317_Diff__insert0,axiom,
    ! [Xc: $o,A: set_o,B: set_o] :
      ( ~ aa_set_o_o(member_o((Xc)),A)
     => ( minus_minus_set_o(A,insert_o((Xc),B)) = minus_minus_set_o(A,B) ) ) ).

% Diff_insert0
tff(fact_1318_Diff__insert0,axiom,
    ! [Xc: real,A: set_real,B: set_real] :
      ( ~ aa_set_real_o(member_real(Xc),A)
     => ( minus_minus_set_real(A,insert_real(Xc,B)) = minus_minus_set_real(A,B) ) ) ).

% Diff_insert0
tff(fact_1319_Diff__insert0,axiom,
    ! [Xc: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),A)
     => ( minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xc,B)) = minus_5127226145743854075T_VEBT(A,B) ) ) ).

% Diff_insert0
tff(fact_1320_Diff__insert0,axiom,
    ! [Xc: int,A: set_int,B: set_int] :
      ( ~ aa_set_int_o(member_int(Xc),A)
     => ( minus_minus_set_int(A,insert_int(Xc,B)) = minus_minus_set_int(A,B) ) ) ).

% Diff_insert0
tff(fact_1321_Diff__insert0,axiom,
    ! [Xc: complex,A: set_complex,B: set_complex] :
      ( ~ aa_set_complex_o(member_complex(Xc),A)
     => ( minus_811609699411566653omplex(A,insert_complex(Xc,B)) = minus_811609699411566653omplex(A,B) ) ) ).

% Diff_insert0
tff(fact_1322_Diff__insert0,axiom,
    ! [Xc: nat,A: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xc),A)
     => ( minus_minus_set_nat(A,insert_nat(Xc,B)) = minus_minus_set_nat(A,B) ) ) ).

% Diff_insert0
tff(fact_1323_psubsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( ( A != B )
       => aa_set_nat_o(ord_less_set_nat(A),B) ) ) ).

% psubsetI
tff(fact_1324_merge__pure__star,axiom,
    ! [A2: $o,B2: $o] :
      aa_assn_assn(times_times_assn(aa_o_assn(pure_assn,(A2))),aa_o_assn(pure_assn,(B2))) = aa_o_assn(pure_assn,
        ( (A2)
        & (B2) )) ).

% merge_pure_star
tff(fact_1325_le__add__diff__inverse2,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1326_le__add__diff__inverse2,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1327_le__add__diff__inverse2,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1328_le__add__diff__inverse2,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A2),B2)),B2) = A2 ) ) ).

% le_add_diff_inverse2
tff(fact_1329_le__add__diff__inverse,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1330_le__add__diff__inverse,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_rat(plus_plus_rat(B2),aa_rat_rat(minus_minus_rat(A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1331_le__add__diff__inverse,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1332_le__add__diff__inverse,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_int(plus_plus_int(B2),aa_int_int(minus_minus_int(A2),B2)) = A2 ) ) ).

% le_add_diff_inverse
tff(fact_1333_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A2),C2)),aa_complex_complex(times_times_complex(C2),B2)) = aa_complex_complex(divide1717551699836669952omplex(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1334_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1335_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(C2),B2)) = aa_rat_rat(divide_divide_rat(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1336_nonzero__mult__div__cancel__right,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1337_nonzero__mult__div__cancel__right,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1338_nonzero__mult__div__cancel__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1339_nonzero__mult__div__cancel__right,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1340_nonzero__mult__div__cancel__right,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A2),B2)),B2) = A2 ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1341_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A2),C2)),aa_complex_complex(times_times_complex(B2),C2)) = aa_complex_complex(divide1717551699836669952omplex(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1342_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1343_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) = aa_rat_rat(divide_divide_rat(A2),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1344_singleton__insert__inj__eq_H,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( ( insert_VEBT_VEBT(A2,A) = insert_VEBT_VEBT(B2,bot_bo8194388402131092736T_VEBT) )
    <=> ( ( A2 = B2 )
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(B2,bot_bo8194388402131092736T_VEBT)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1345_singleton__insert__inj__eq_H,axiom,
    ! [A2: real,A: set_real,B2: real] :
      ( ( insert_real(A2,A) = insert_real(B2,bot_bot_set_real) )
    <=> ( ( A2 = B2 )
        & aa_set_real_o(ord_less_eq_set_real(A),insert_real(B2,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1346_singleton__insert__inj__eq_H,axiom,
    ! [A2: $o,A: set_o,B2: $o] :
      ( ( insert_o((A2),A) = insert_o((B2),bot_bot_set_o) )
    <=> ( ( (A2)
        <=> (B2) )
        & aa_set_o_o(ord_less_eq_set_o(A),insert_o((B2),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1347_singleton__insert__inj__eq_H,axiom,
    ! [A2: int,A: set_int,B2: int] :
      ( ( insert_int(A2,A) = insert_int(B2,bot_bot_set_int) )
    <=> ( ( A2 = B2 )
        & aa_set_int_o(ord_less_eq_set_int(A),insert_int(B2,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1348_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B2: nat] :
      ( ( insert_nat(A2,A) = insert_nat(B2,bot_bot_set_nat) )
    <=> ( ( A2 = B2 )
        & aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(B2,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq'
tff(fact_1349_singleton__insert__inj__eq,axiom,
    ! [B2: vEBT_VEBT,A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT(B2,bot_bo8194388402131092736T_VEBT) = insert_VEBT_VEBT(A2,A) )
    <=> ( ( A2 = B2 )
        & aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(B2,bot_bo8194388402131092736T_VEBT)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1350_singleton__insert__inj__eq,axiom,
    ! [B2: real,A2: real,A: set_real] :
      ( ( insert_real(B2,bot_bot_set_real) = insert_real(A2,A) )
    <=> ( ( A2 = B2 )
        & aa_set_real_o(ord_less_eq_set_real(A),insert_real(B2,bot_bot_set_real)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1351_singleton__insert__inj__eq,axiom,
    ! [B2: $o,A2: $o,A: set_o] :
      ( ( insert_o((B2),bot_bot_set_o) = insert_o((A2),A) )
    <=> ( ( (A2)
        <=> (B2) )
        & aa_set_o_o(ord_less_eq_set_o(A),insert_o((B2),bot_bot_set_o)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1352_singleton__insert__inj__eq,axiom,
    ! [B2: int,A2: int,A: set_int] :
      ( ( insert_int(B2,bot_bot_set_int) = insert_int(A2,A) )
    <=> ( ( A2 = B2 )
        & aa_set_int_o(ord_less_eq_set_int(A),insert_int(B2,bot_bot_set_int)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1353_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A2: nat,A: set_nat] :
      ( ( insert_nat(B2,bot_bot_set_nat) = insert_nat(A2,A) )
    <=> ( ( A2 = B2 )
        & aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(B2,bot_bot_set_nat)) ) ) ).

% singleton_insert_inj_eq
tff(fact_1354_Diff__eq__empty__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( minus_minus_set_real(A,B) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(A),B) ) ).

% Diff_eq_empty_iff
tff(fact_1355_Diff__eq__empty__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( minus_minus_set_o(A,B) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(A),B) ) ).

% Diff_eq_empty_iff
tff(fact_1356_Diff__eq__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( minus_minus_set_int(A,B) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(A),B) ) ).

% Diff_eq_empty_iff
tff(fact_1357_Diff__eq__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( minus_minus_set_nat(A,B) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% Diff_eq_empty_iff
tff(fact_1358_in__mono,axiom,
    ! [A: set_real,B: set_real,Xc: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B)
     => ( aa_set_real_o(member_real(Xc),A)
       => aa_set_real_o(member_real(Xc),B) ) ) ).

% in_mono
tff(fact_1359_in__mono,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,Xc: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),A)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),B) ) ) ).

% in_mono
tff(fact_1360_in__mono,axiom,
    ! [A: set_int,B: set_int,Xc: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(member_int(Xc),A)
       => aa_set_int_o(member_int(Xc),B) ) ) ).

% in_mono
tff(fact_1361_in__mono,axiom,
    ! [A: set_complex,B: set_complex,Xc: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B)
     => ( aa_set_complex_o(member_complex(Xc),A)
       => aa_set_complex_o(member_complex(Xc),B) ) ) ).

% in_mono
tff(fact_1362_in__mono,axiom,
    ! [A: set_nat,B: set_nat,Xc: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(member_nat(Xc),A)
       => aa_set_nat_o(member_nat(Xc),B) ) ) ).

% in_mono
tff(fact_1363_subsetD,axiom,
    ! [A: set_real,B: set_real,C2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B)
     => ( aa_set_real_o(member_real(C2),A)
       => aa_set_real_o(member_real(C2),B) ) ) ).

% subsetD
tff(fact_1364_subsetD,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,C2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B) ) ) ).

% subsetD
tff(fact_1365_subsetD,axiom,
    ! [A: set_int,B: set_int,C2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(member_int(C2),A)
       => aa_set_int_o(member_int(C2),B) ) ) ).

% subsetD
tff(fact_1366_subsetD,axiom,
    ! [A: set_complex,B: set_complex,C2: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B)
     => ( aa_set_complex_o(member_complex(C2),A)
       => aa_set_complex_o(member_complex(C2),B) ) ) ).

% subsetD
tff(fact_1367_subsetD,axiom,
    ! [A: set_nat,B: set_nat,C2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(member_nat(C2),A)
       => aa_set_nat_o(member_nat(C2),B) ) ) ).

% subsetD
tff(fact_1368_psubsetD,axiom,
    ! [A: set_nat,B: set_nat,C2: nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ( aa_set_nat_o(member_nat(C2),A)
       => aa_set_nat_o(member_nat(C2),B) ) ) ).

% psubsetD
tff(fact_1369_psubsetD,axiom,
    ! [A: set_real,B: set_real,C2: real] :
      ( aa_set_real_o(ord_less_set_real(A),B)
     => ( aa_set_real_o(member_real(C2),A)
       => aa_set_real_o(member_real(C2),B) ) ) ).

% psubsetD
tff(fact_1370_psubsetD,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,C2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A),B)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A)
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B) ) ) ).

% psubsetD
tff(fact_1371_psubsetD,axiom,
    ! [A: set_int,B: set_int,C2: int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
     => ( aa_set_int_o(member_int(C2),A)
       => aa_set_int_o(member_int(C2),B) ) ) ).

% psubsetD
tff(fact_1372_psubsetD,axiom,
    ! [A: set_complex,B: set_complex,C2: complex] :
      ( aa_set_complex_o(ord_less_set_complex(A),B)
     => ( aa_set_complex_o(member_complex(C2),A)
       => aa_set_complex_o(member_complex(C2),B) ) ) ).

% psubsetD
tff(fact_1373_psubsetE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
         => aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).

% psubsetE
tff(fact_1374_equalityE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ~ ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
         => ~ aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).

% equalityE
tff(fact_1375_subset__eq,axiom,
    ! [A: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B)
    <=> ! [X2: real] :
          ( aa_set_real_o(member_real(X2),A)
         => aa_set_real_o(member_real(X2),B) ) ) ).

% subset_eq
tff(fact_1376_subset__eq,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B)
    <=> ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),B) ) ) ).

% subset_eq
tff(fact_1377_subset__eq,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
    <=> ! [X2: int] :
          ( aa_set_int_o(member_int(X2),A)
         => aa_set_int_o(member_int(X2),B) ) ) ).

% subset_eq
tff(fact_1378_subset__eq,axiom,
    ! [A: set_complex,B: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B)
    <=> ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),A)
         => aa_set_complex_o(member_complex(X2),B) ) ) ).

% subset_eq
tff(fact_1379_subset__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),A)
         => aa_set_nat_o(member_nat(X2),B) ) ) ).

% subset_eq
tff(fact_1380_equalityD1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% equalityD1
tff(fact_1381_Set_OequalityD2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ).

% Set.equalityD2
tff(fact_1382_psubset__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & ( A != B ) ) ) ).

% psubset_eq
tff(fact_1383_subset__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B)
    <=> ! [T: real] :
          ( aa_set_real_o(member_real(T),A)
         => aa_set_real_o(member_real(T),B) ) ) ).

% subset_iff
tff(fact_1384_subset__iff,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B)
    <=> ! [T: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T),A)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(T),B) ) ) ).

% subset_iff
tff(fact_1385_subset__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
    <=> ! [T: int] :
          ( aa_set_int_o(member_int(T),A)
         => aa_set_int_o(member_int(T),B) ) ) ).

% subset_iff
tff(fact_1386_subset__iff,axiom,
    ! [A: set_complex,B: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B)
    <=> ! [T: complex] :
          ( aa_set_complex_o(member_complex(T),A)
         => aa_set_complex_o(member_complex(T),B) ) ) ).

% subset_iff
tff(fact_1387_subset__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ! [T: nat] :
          ( aa_set_nat_o(member_nat(T),A)
         => aa_set_nat_o(member_nat(T),B) ) ) ).

% subset_iff
tff(fact_1388_subset__refl,axiom,
    ! [A: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A),A) ).

% subset_refl
tff(fact_1389_Collect__mono,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( aa_int_o(P,X3)
         => aa_int_o(Q,X3) )
     => aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q)) ) ).

% Collect_mono
tff(fact_1390_Collect__mono,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X3: complex] :
          ( aa_complex_o(P,X3)
         => aa_complex_o(Q,X3) )
     => aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q)) ) ).

% Collect_mono
tff(fact_1391_Collect__mono,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ! [X3: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X3)
         => aa_Pro4592068669816069595_int_o(Q,X3) )
     => ord_le2843351958646193337nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q)) ) ).

% Collect_mono
tff(fact_1392_Collect__mono,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(P,X3)
         => aa_nat_o(Q,X3) )
     => aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q)) ) ).

% Collect_mono
tff(fact_1393_subset__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_eq_set_nat(A),C) ) ) ).

% subset_trans
tff(fact_1394_set__eq__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).

% set_eq_subset
tff(fact_1395_Collect__mono__iff,axiom,
    ! [P: fun_int_o,Q: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(P)),collect_int(Q))
    <=> ! [X2: int] :
          ( aa_int_o(P,X2)
         => aa_int_o(Q,X2) ) ) ).

% Collect_mono_iff
tff(fact_1396_Collect__mono__iff,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(P)),collect_complex(Q))
    <=> ! [X2: complex] :
          ( aa_complex_o(P,X2)
         => aa_complex_o(Q,X2) ) ) ).

% Collect_mono_iff
tff(fact_1397_Collect__mono__iff,axiom,
    ! [P: fun_Pr3238374608701176162_int_o,Q: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(collec213857154873943460nt_int(P),collec213857154873943460nt_int(Q))
    <=> ! [X2: product_prod_int_int] :
          ( aa_Pro4592068669816069595_int_o(P,X2)
         => aa_Pro4592068669816069595_int_o(Q,X2) ) ) ).

% Collect_mono_iff
tff(fact_1398_Collect__mono__iff,axiom,
    ! [P: fun_nat_o,Q: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(P)),collect_nat(Q))
    <=> ! [X2: nat] :
          ( aa_nat_o(P,X2)
         => aa_nat_o(Q,X2) ) ) ).

% Collect_mono_iff
tff(fact_1399_psubset__imp__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ).

% psubset_imp_subset
tff(fact_1400_subset__Collect__conv,axiom,
    ! [S: set_int,P: fun_int_o] :
      ( aa_set_int_o(ord_less_eq_set_int(S),collect_int(P))
    <=> ! [X2: int] :
          ( aa_set_int_o(member_int(X2),S)
         => aa_int_o(P,X2) ) ) ).

% subset_Collect_conv
tff(fact_1401_subset__Collect__conv,axiom,
    ! [S: set_complex,P: fun_complex_o] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S),collect_complex(P))
    <=> ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),S)
         => aa_complex_o(P,X2) ) ) ).

% subset_Collect_conv
tff(fact_1402_subset__Collect__conv,axiom,
    ! [S: set_Pr958786334691620121nt_int,P: fun_Pr3238374608701176162_int_o] :
      ( ord_le2843351958646193337nt_int(S,collec213857154873943460nt_int(P))
    <=> ! [X2: product_prod_int_int] :
          ( member5262025264175285858nt_int(X2,S)
         => aa_Pro4592068669816069595_int_o(P,X2) ) ) ).

% subset_Collect_conv
tff(fact_1403_subset__Collect__conv,axiom,
    ! [S: set_nat,P: fun_nat_o] :
      ( aa_set_nat_o(ord_less_eq_set_nat(S),collect_nat(P))
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),S)
         => aa_nat_o(P,X2) ) ) ).

% subset_Collect_conv
tff(fact_1404_psubset__subset__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% psubset_subset_trans
tff(fact_1405_subset__not__subset__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ) ).

% subset_not_subset_eq
tff(fact_1406_subset__psubset__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_set_nat(B),C)
       => aa_set_nat_o(ord_less_set_nat(A),C) ) ) ).

% subset_psubset_trans
tff(fact_1407_subset__iff__psubset__eq,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ( aa_set_nat_o(ord_less_set_nat(A),B)
        | ( A = B ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1408_diff__commute,axiom,
    ! [I: nat,J2: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J2)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J2) ).

% diff_commute
tff(fact_1409_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xc: produc9217583074051984497at_nat] :
      ~ ! [F3: fun_nat_fun_nat_nat,A3: nat,B3: nat,Acc: nat] : Xc != produc2818902965508115497at_nat(F3,produc487386426758144856at_nat(A3,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1410_linorder__neqE__linordered__idom,axiom,
    ! [Xc: real,Ya: real] :
      ( ( Xc != Ya )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1411_linorder__neqE__linordered__idom,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ( Xc != Ya )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1412_linorder__neqE__linordered__idom,axiom,
    ! [Xc: int,Ya: int] :
      ( ( Xc != Ya )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_1413_linordered__field__no__ub,axiom,
    ! [X4: real] :
    ? [X_12: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X4),X_12) ).

% linordered_field_no_ub
tff(fact_1414_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_12: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),X_12) ).

% linordered_field_no_ub
tff(fact_1415_linordered__field__no__lb,axiom,
    ! [X4: real] :
    ? [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),X4) ).

% linordered_field_no_lb
tff(fact_1416_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),X4) ).

% linordered_field_no_lb
tff(fact_1417_realpow__pos__nth2,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ? [R2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R2),aa_nat_nat(suc,Nb)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_1418_not__psubset__empty,axiom,
    ! [A: set_real] : ~ aa_set_real_o(ord_less_set_real(A),bot_bot_set_real) ).

% not_psubset_empty
tff(fact_1419_not__psubset__empty,axiom,
    ! [A: set_o] : ~ aa_set_o_o(ord_less_set_o(A),bot_bot_set_o) ).

% not_psubset_empty
tff(fact_1420_not__psubset__empty,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A),bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_1421_not__psubset__empty,axiom,
    ! [A: set_int] : ~ aa_set_int_o(ord_less_set_int(A),bot_bot_set_int) ).

% not_psubset_empty
tff(fact_1422_ex__in__conv,axiom,
    ! [A: set_VEBT_VEBT] :
      ( ? [X2: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),A)
    <=> ( A != bot_bo8194388402131092736T_VEBT ) ) ).

% ex_in_conv
tff(fact_1423_ex__in__conv,axiom,
    ! [A: set_complex] :
      ( ? [X2: complex] : aa_set_complex_o(member_complex(X2),A)
    <=> ( A != bot_bot_set_complex ) ) ).

% ex_in_conv
tff(fact_1424_ex__in__conv,axiom,
    ! [A: set_real] :
      ( ? [X2: real] : aa_set_real_o(member_real(X2),A)
    <=> ( A != bot_bot_set_real ) ) ).

% ex_in_conv
tff(fact_1425_ex__in__conv,axiom,
    ! [A: set_o] :
      ( ? [X2: $o] : aa_set_o_o(member_o((X2)),A)
    <=> ( A != bot_bot_set_o ) ) ).

% ex_in_conv
tff(fact_1426_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ? [X2: nat] : aa_set_nat_o(member_nat(X2),A)
    <=> ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
tff(fact_1427_ex__in__conv,axiom,
    ! [A: set_int] :
      ( ? [X2: int] : aa_set_int_o(member_int(X2),A)
    <=> ( A != bot_bot_set_int ) ) ).

% ex_in_conv
tff(fact_1428_equals0I,axiom,
    ! [A: set_VEBT_VEBT] :
      ( ! [Y: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y),A)
     => ( A = bot_bo8194388402131092736T_VEBT ) ) ).

% equals0I
tff(fact_1429_equals0I,axiom,
    ! [A: set_complex] :
      ( ! [Y: complex] : ~ aa_set_complex_o(member_complex(Y),A)
     => ( A = bot_bot_set_complex ) ) ).

% equals0I
tff(fact_1430_equals0I,axiom,
    ! [A: set_real] :
      ( ! [Y: real] : ~ aa_set_real_o(member_real(Y),A)
     => ( A = bot_bot_set_real ) ) ).

% equals0I
tff(fact_1431_equals0I,axiom,
    ! [A: set_o] :
      ( ! [Y: $o] : ~ aa_set_o_o(member_o((Y)),A)
     => ( A = bot_bot_set_o ) ) ).

% equals0I
tff(fact_1432_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y: nat] : ~ aa_set_nat_o(member_nat(Y),A)
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
tff(fact_1433_equals0I,axiom,
    ! [A: set_int] :
      ( ! [Y: int] : ~ aa_set_int_o(member_int(Y),A)
     => ( A = bot_bot_set_int ) ) ).

% equals0I
tff(fact_1434_equals0D,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( ( A = bot_bo8194388402131092736T_VEBT )
     => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),A) ) ).

% equals0D
tff(fact_1435_equals0D,axiom,
    ! [A: set_complex,A2: complex] :
      ( ( A = bot_bot_set_complex )
     => ~ aa_set_complex_o(member_complex(A2),A) ) ).

% equals0D
tff(fact_1436_equals0D,axiom,
    ! [A: set_real,A2: real] :
      ( ( A = bot_bot_set_real )
     => ~ aa_set_real_o(member_real(A2),A) ) ).

% equals0D
tff(fact_1437_equals0D,axiom,
    ! [A: set_o,A2: $o] :
      ( ( A = bot_bot_set_o )
     => ~ aa_set_o_o(member_o((A2)),A) ) ).

% equals0D
tff(fact_1438_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ aa_set_nat_o(member_nat(A2),A) ) ).

% equals0D
tff(fact_1439_equals0D,axiom,
    ! [A: set_int,A2: int] :
      ( ( A = bot_bot_set_int )
     => ~ aa_set_int_o(member_int(A2),A) ) ).

% equals0D
tff(fact_1440_emptyE,axiom,
    ! [A2: vEBT_VEBT] : ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),bot_bo8194388402131092736T_VEBT) ).

% emptyE
tff(fact_1441_emptyE,axiom,
    ! [A2: complex] : ~ aa_set_complex_o(member_complex(A2),bot_bot_set_complex) ).

% emptyE
tff(fact_1442_emptyE,axiom,
    ! [A2: real] : ~ aa_set_real_o(member_real(A2),bot_bot_set_real) ).

% emptyE
tff(fact_1443_emptyE,axiom,
    ! [A2: $o] : ~ aa_set_o_o(member_o((A2)),bot_bot_set_o) ).

% emptyE
tff(fact_1444_emptyE,axiom,
    ! [A2: nat] : ~ aa_set_nat_o(member_nat(A2),bot_bot_set_nat) ).

% emptyE
tff(fact_1445_emptyE,axiom,
    ! [A2: int] : ~ aa_set_int_o(member_int(A2),bot_bot_set_int) ).

% emptyE
tff(fact_1446_mk__disjoint__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(member_o((A2)),A)
     => ? [B5: set_o] :
          ( ( A = insert_o((A2),B5) )
          & ~ aa_set_o_o(member_o((A2)),B5) ) ) ).

% mk_disjoint_insert
tff(fact_1447_mk__disjoint__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ? [B5: set_nat] :
          ( ( A = insert_nat(A2,B5) )
          & ~ aa_set_nat_o(member_nat(A2),B5) ) ) ).

% mk_disjoint_insert
tff(fact_1448_mk__disjoint__insert,axiom,
    ! [A2: real,A: set_real] :
      ( aa_set_real_o(member_real(A2),A)
     => ? [B5: set_real] :
          ( ( A = insert_real(A2,B5) )
          & ~ aa_set_real_o(member_real(A2),B5) ) ) ).

% mk_disjoint_insert
tff(fact_1449_mk__disjoint__insert,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),A)
     => ? [B5: set_VEBT_VEBT] :
          ( ( A = insert_VEBT_VEBT(A2,B5) )
          & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),B5) ) ) ).

% mk_disjoint_insert
tff(fact_1450_mk__disjoint__insert,axiom,
    ! [A2: int,A: set_int] :
      ( aa_set_int_o(member_int(A2),A)
     => ? [B5: set_int] :
          ( ( A = insert_int(A2,B5) )
          & ~ aa_set_int_o(member_int(A2),B5) ) ) ).

% mk_disjoint_insert
tff(fact_1451_mk__disjoint__insert,axiom,
    ! [A2: complex,A: set_complex] :
      ( aa_set_complex_o(member_complex(A2),A)
     => ? [B5: set_complex] :
          ( ( A = insert_complex(A2,B5) )
          & ~ aa_set_complex_o(member_complex(A2),B5) ) ) ).

% mk_disjoint_insert
tff(fact_1452_insert__commute,axiom,
    ! [Xc: nat,Ya: nat,A: set_nat] : insert_nat(Xc,insert_nat(Ya,A)) = insert_nat(Ya,insert_nat(Xc,A)) ).

% insert_commute
tff(fact_1453_insert__commute,axiom,
    ! [Xc: int,Ya: int,A: set_int] : insert_int(Xc,insert_int(Ya,A)) = insert_int(Ya,insert_int(Xc,A)) ).

% insert_commute
tff(fact_1454_insert__commute,axiom,
    ! [Xc: vEBT_VEBT,Ya: vEBT_VEBT,A: set_VEBT_VEBT] : insert_VEBT_VEBT(Xc,insert_VEBT_VEBT(Ya,A)) = insert_VEBT_VEBT(Ya,insert_VEBT_VEBT(Xc,A)) ).

% insert_commute
tff(fact_1455_insert__commute,axiom,
    ! [Xc: real,Ya: real,A: set_real] : insert_real(Xc,insert_real(Ya,A)) = insert_real(Ya,insert_real(Xc,A)) ).

% insert_commute
tff(fact_1456_insert__commute,axiom,
    ! [Xc: $o,Ya: $o,A: set_o] : insert_o((Xc),insert_o((Ya),A)) = insert_o((Ya),insert_o((Xc),A)) ).

% insert_commute
tff(fact_1457_insert__eq__iff,axiom,
    ! [A2: $o,A: set_o,B2: $o,B: set_o] :
      ( ~ aa_set_o_o(member_o((A2)),A)
     => ( ~ aa_set_o_o(member_o((B2)),B)
       => ( ( insert_o((A2),A) = insert_o((B2),B) )
        <=> $ite(
              (A2) = (B2),
              A = B,
              ? [C3: set_o] :
                ( ( A = insert_o((B2),C3) )
                & ~ aa_set_o_o(member_o((B2)),C3)
                & ( B = insert_o((A2),C3) )
                & ~ aa_set_o_o(member_o((A2)),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1458_insert__eq__iff,axiom,
    ! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
      ( ~ aa_set_nat_o(member_nat(A2),A)
     => ( ~ aa_set_nat_o(member_nat(B2),B)
       => ( ( insert_nat(A2,A) = insert_nat(B2,B) )
        <=> $ite(
              A2 = B2,
              A = B,
              ? [C3: set_nat] :
                ( ( A = insert_nat(B2,C3) )
                & ~ aa_set_nat_o(member_nat(B2),C3)
                & ( B = insert_nat(A2,C3) )
                & ~ aa_set_nat_o(member_nat(A2),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1459_insert__eq__iff,axiom,
    ! [A2: real,A: set_real,B2: real,B: set_real] :
      ( ~ aa_set_real_o(member_real(A2),A)
     => ( ~ aa_set_real_o(member_real(B2),B)
       => ( ( insert_real(A2,A) = insert_real(B2,B) )
        <=> $ite(
              A2 = B2,
              A = B,
              ? [C3: set_real] :
                ( ( A = insert_real(B2,C3) )
                & ~ aa_set_real_o(member_real(B2),C3)
                & ( B = insert_real(A2,C3) )
                & ~ aa_set_real_o(member_real(A2),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1460_insert__eq__iff,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT,B2: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),A)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),B)
       => ( ( insert_VEBT_VEBT(A2,A) = insert_VEBT_VEBT(B2,B) )
        <=> $ite(
              A2 = B2,
              A = B,
              ? [C3: set_VEBT_VEBT] :
                ( ( A = insert_VEBT_VEBT(B2,C3) )
                & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),C3)
                & ( B = insert_VEBT_VEBT(A2,C3) )
                & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1461_insert__eq__iff,axiom,
    ! [A2: int,A: set_int,B2: int,B: set_int] :
      ( ~ aa_set_int_o(member_int(A2),A)
     => ( ~ aa_set_int_o(member_int(B2),B)
       => ( ( insert_int(A2,A) = insert_int(B2,B) )
        <=> $ite(
              A2 = B2,
              A = B,
              ? [C3: set_int] :
                ( ( A = insert_int(B2,C3) )
                & ~ aa_set_int_o(member_int(B2),C3)
                & ( B = insert_int(A2,C3) )
                & ~ aa_set_int_o(member_int(A2),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1462_insert__eq__iff,axiom,
    ! [A2: complex,A: set_complex,B2: complex,B: set_complex] :
      ( ~ aa_set_complex_o(member_complex(A2),A)
     => ( ~ aa_set_complex_o(member_complex(B2),B)
       => ( ( insert_complex(A2,A) = insert_complex(B2,B) )
        <=> $ite(
              A2 = B2,
              A = B,
              ? [C3: set_complex] :
                ( ( A = insert_complex(B2,C3) )
                & ~ aa_set_complex_o(member_complex(B2),C3)
                & ( B = insert_complex(A2,C3) )
                & ~ aa_set_complex_o(member_complex(A2),C3) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_1463_insert__absorb,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( insert_o((A2),A) = A ) ) ).

% insert_absorb
tff(fact_1464_insert__absorb,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( insert_nat(A2,A) = A ) ) ).

% insert_absorb
tff(fact_1465_insert__absorb,axiom,
    ! [A2: real,A: set_real] :
      ( aa_set_real_o(member_real(A2),A)
     => ( insert_real(A2,A) = A ) ) ).

% insert_absorb
tff(fact_1466_insert__absorb,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),A)
     => ( insert_VEBT_VEBT(A2,A) = A ) ) ).

% insert_absorb
tff(fact_1467_insert__absorb,axiom,
    ! [A2: int,A: set_int] :
      ( aa_set_int_o(member_int(A2),A)
     => ( insert_int(A2,A) = A ) ) ).

% insert_absorb
tff(fact_1468_insert__absorb,axiom,
    ! [A2: complex,A: set_complex] :
      ( aa_set_complex_o(member_complex(A2),A)
     => ( insert_complex(A2,A) = A ) ) ).

% insert_absorb
tff(fact_1469_insert__ident,axiom,
    ! [Xc: $o,A: set_o,B: set_o] :
      ( ~ aa_set_o_o(member_o((Xc)),A)
     => ( ~ aa_set_o_o(member_o((Xc)),B)
       => ( ( insert_o((Xc),A) = insert_o((Xc),B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_1470_insert__ident,axiom,
    ! [Xc: nat,A: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xc),A)
     => ( ~ aa_set_nat_o(member_nat(Xc),B)
       => ( ( insert_nat(Xc,A) = insert_nat(Xc,B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_1471_insert__ident,axiom,
    ! [Xc: real,A: set_real,B: set_real] :
      ( ~ aa_set_real_o(member_real(Xc),A)
     => ( ~ aa_set_real_o(member_real(Xc),B)
       => ( ( insert_real(Xc,A) = insert_real(Xc,B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_1472_insert__ident,axiom,
    ! [Xc: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),A)
     => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),B)
       => ( ( insert_VEBT_VEBT(Xc,A) = insert_VEBT_VEBT(Xc,B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_1473_insert__ident,axiom,
    ! [Xc: int,A: set_int,B: set_int] :
      ( ~ aa_set_int_o(member_int(Xc),A)
     => ( ~ aa_set_int_o(member_int(Xc),B)
       => ( ( insert_int(Xc,A) = insert_int(Xc,B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_1474_insert__ident,axiom,
    ! [Xc: complex,A: set_complex,B: set_complex] :
      ( ~ aa_set_complex_o(member_complex(Xc),A)
     => ( ~ aa_set_complex_o(member_complex(Xc),B)
       => ( ( insert_complex(Xc,A) = insert_complex(Xc,B) )
        <=> ( A = B ) ) ) ) ).

% insert_ident
tff(fact_1475_Set_Oset__insert,axiom,
    ! [Xc: $o,A: set_o] :
      ( aa_set_o_o(member_o((Xc)),A)
     => ~ ! [B5: set_o] :
            ( ( A = insert_o((Xc),B5) )
           => aa_set_o_o(member_o((Xc)),B5) ) ) ).

% Set.set_insert
tff(fact_1476_Set_Oset__insert,axiom,
    ! [Xc: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(Xc),A)
     => ~ ! [B5: set_nat] :
            ( ( A = insert_nat(Xc,B5) )
           => aa_set_nat_o(member_nat(Xc),B5) ) ) ).

% Set.set_insert
tff(fact_1477_Set_Oset__insert,axiom,
    ! [Xc: real,A: set_real] :
      ( aa_set_real_o(member_real(Xc),A)
     => ~ ! [B5: set_real] :
            ( ( A = insert_real(Xc,B5) )
           => aa_set_real_o(member_real(Xc),B5) ) ) ).

% Set.set_insert
tff(fact_1478_Set_Oset__insert,axiom,
    ! [Xc: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),A)
     => ~ ! [B5: set_VEBT_VEBT] :
            ( ( A = insert_VEBT_VEBT(Xc,B5) )
           => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),B5) ) ) ).

% Set.set_insert
tff(fact_1479_Set_Oset__insert,axiom,
    ! [Xc: int,A: set_int] :
      ( aa_set_int_o(member_int(Xc),A)
     => ~ ! [B5: set_int] :
            ( ( A = insert_int(Xc,B5) )
           => aa_set_int_o(member_int(Xc),B5) ) ) ).

% Set.set_insert
tff(fact_1480_Set_Oset__insert,axiom,
    ! [Xc: complex,A: set_complex] :
      ( aa_set_complex_o(member_complex(Xc),A)
     => ~ ! [B5: set_complex] :
            ( ( A = insert_complex(Xc,B5) )
           => aa_set_complex_o(member_complex(Xc),B5) ) ) ).

% Set.set_insert
tff(fact_1481_insertI2,axiom,
    ! [A2: $o,B: set_o,B2: $o] :
      ( aa_set_o_o(member_o((A2)),B)
     => aa_set_o_o(member_o((A2)),insert_o((B2),B)) ) ).

% insertI2
tff(fact_1482_insertI2,axiom,
    ! [A2: nat,B: set_nat,B2: nat] :
      ( aa_set_nat_o(member_nat(A2),B)
     => aa_set_nat_o(member_nat(A2),insert_nat(B2,B)) ) ).

% insertI2
tff(fact_1483_insertI2,axiom,
    ! [A2: real,B: set_real,B2: real] :
      ( aa_set_real_o(member_real(A2),B)
     => aa_set_real_o(member_real(A2),insert_real(B2,B)) ) ).

% insertI2
tff(fact_1484_insertI2,axiom,
    ! [A2: vEBT_VEBT,B: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),B)
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),insert_VEBT_VEBT(B2,B)) ) ).

% insertI2
tff(fact_1485_insertI2,axiom,
    ! [A2: int,B: set_int,B2: int] :
      ( aa_set_int_o(member_int(A2),B)
     => aa_set_int_o(member_int(A2),insert_int(B2,B)) ) ).

% insertI2
tff(fact_1486_insertI2,axiom,
    ! [A2: complex,B: set_complex,B2: complex] :
      ( aa_set_complex_o(member_complex(A2),B)
     => aa_set_complex_o(member_complex(A2),insert_complex(B2,B)) ) ).

% insertI2
tff(fact_1487_insertI1,axiom,
    ! [A2: $o,B: set_o] : aa_set_o_o(member_o((A2)),insert_o((A2),B)) ).

% insertI1
tff(fact_1488_insertI1,axiom,
    ! [A2: nat,B: set_nat] : aa_set_nat_o(member_nat(A2),insert_nat(A2,B)) ).

% insertI1
tff(fact_1489_insertI1,axiom,
    ! [A2: real,B: set_real] : aa_set_real_o(member_real(A2),insert_real(A2,B)) ).

% insertI1
tff(fact_1490_insertI1,axiom,
    ! [A2: vEBT_VEBT,B: set_VEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),insert_VEBT_VEBT(A2,B)) ).

% insertI1
tff(fact_1491_insertI1,axiom,
    ! [A2: int,B: set_int] : aa_set_int_o(member_int(A2),insert_int(A2,B)) ).

% insertI1
tff(fact_1492_insertI1,axiom,
    ! [A2: complex,B: set_complex] : aa_set_complex_o(member_complex(A2),insert_complex(A2,B)) ).

% insertI1
tff(fact_1493_insertE,axiom,
    ! [A2: $o,B2: $o,A: set_o] :
      ( aa_set_o_o(member_o((A2)),insert_o((B2),A))
     => ( ( (A2)
        <=> ~ (B2) )
       => aa_set_o_o(member_o((A2)),A) ) ) ).

% insertE
tff(fact_1494_insertE,axiom,
    ! [A2: nat,B2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),insert_nat(B2,A))
     => ( ( A2 != B2 )
       => aa_set_nat_o(member_nat(A2),A) ) ) ).

% insertE
tff(fact_1495_insertE,axiom,
    ! [A2: real,B2: real,A: set_real] :
      ( aa_set_real_o(member_real(A2),insert_real(B2,A))
     => ( ( A2 != B2 )
       => aa_set_real_o(member_real(A2),A) ) ) ).

% insertE
tff(fact_1496_insertE,axiom,
    ! [A2: vEBT_VEBT,B2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),insert_VEBT_VEBT(B2,A))
     => ( ( A2 != B2 )
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),A) ) ) ).

% insertE
tff(fact_1497_insertE,axiom,
    ! [A2: int,B2: int,A: set_int] :
      ( aa_set_int_o(member_int(A2),insert_int(B2,A))
     => ( ( A2 != B2 )
       => aa_set_int_o(member_int(A2),A) ) ) ).

% insertE
tff(fact_1498_insertE,axiom,
    ! [A2: complex,B2: complex,A: set_complex] :
      ( aa_set_complex_o(member_complex(A2),insert_complex(B2,A))
     => ( ( A2 != B2 )
       => aa_set_complex_o(member_complex(A2),A) ) ) ).

% insertE
tff(fact_1499_subset__insertI2,axiom,
    ! [A: set_int,B: set_int,B2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => aa_set_int_o(ord_less_eq_set_int(A),insert_int(B2,B)) ) ).

% subset_insertI2
tff(fact_1500_subset__insertI2,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(B2,B)) ) ).

% subset_insertI2
tff(fact_1501_subset__insertI2,axiom,
    ! [A: set_real,B: set_real,B2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B)
     => aa_set_real_o(ord_less_eq_set_real(A),insert_real(B2,B)) ) ).

% subset_insertI2
tff(fact_1502_subset__insertI2,axiom,
    ! [A: set_o,B: set_o,B2: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),B)
     => aa_set_o_o(ord_less_eq_set_o(A),insert_o((B2),B)) ) ).

% subset_insertI2
tff(fact_1503_subset__insertI2,axiom,
    ! [A: set_nat,B: set_nat,B2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(B2,B)) ) ).

% subset_insertI2
tff(fact_1504_subset__insertI,axiom,
    ! [B: set_int,A2: int] : aa_set_int_o(ord_less_eq_set_int(B),insert_int(A2,B)) ).

% subset_insertI
tff(fact_1505_subset__insertI,axiom,
    ! [B: set_VEBT_VEBT,A2: vEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(B),insert_VEBT_VEBT(A2,B)) ).

% subset_insertI
tff(fact_1506_subset__insertI,axiom,
    ! [B: set_real,A2: real] : aa_set_real_o(ord_less_eq_set_real(B),insert_real(A2,B)) ).

% subset_insertI
tff(fact_1507_subset__insertI,axiom,
    ! [B: set_o,A2: $o] : aa_set_o_o(ord_less_eq_set_o(B),insert_o((A2),B)) ).

% subset_insertI
tff(fact_1508_subset__insertI,axiom,
    ! [B: set_nat,A2: nat] : aa_set_nat_o(ord_less_eq_set_nat(B),insert_nat(A2,B)) ).

% subset_insertI
tff(fact_1509_subset__insert,axiom,
    ! [Xc: $o,A: set_o,B: set_o] :
      ( ~ aa_set_o_o(member_o((Xc)),A)
     => ( aa_set_o_o(ord_less_eq_set_o(A),insert_o((Xc),B))
      <=> aa_set_o_o(ord_less_eq_set_o(A),B) ) ) ).

% subset_insert
tff(fact_1510_subset__insert,axiom,
    ! [Xc: real,A: set_real,B: set_real] :
      ( ~ aa_set_real_o(member_real(Xc),A)
     => ( aa_set_real_o(ord_less_eq_set_real(A),insert_real(Xc,B))
      <=> aa_set_real_o(ord_less_eq_set_real(A),B) ) ) ).

% subset_insert
tff(fact_1511_subset__insert,axiom,
    ! [Xc: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),A)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(Xc,B))
      <=> aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B) ) ) ).

% subset_insert
tff(fact_1512_subset__insert,axiom,
    ! [Xc: int,A: set_int,B: set_int] :
      ( ~ aa_set_int_o(member_int(Xc),A)
     => ( aa_set_int_o(ord_less_eq_set_int(A),insert_int(Xc,B))
      <=> aa_set_int_o(ord_less_eq_set_int(A),B) ) ) ).

% subset_insert
tff(fact_1513_subset__insert,axiom,
    ! [Xc: complex,A: set_complex,B: set_complex] :
      ( ~ aa_set_complex_o(member_complex(Xc),A)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),insert_complex(Xc,B))
      <=> aa_set_complex_o(ord_le211207098394363844omplex(A),B) ) ) ).

% subset_insert
tff(fact_1514_subset__insert,axiom,
    ! [Xc: nat,A: set_nat,B: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xc),A)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(Xc,B))
      <=> aa_set_nat_o(ord_less_eq_set_nat(A),B) ) ) ).

% subset_insert
tff(fact_1515_insert__mono,axiom,
    ! [C: set_int,D: set_int,A2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(C),D)
     => aa_set_int_o(ord_less_eq_set_int(insert_int(A2,C)),insert_int(A2,D)) ) ).

% insert_mono
tff(fact_1516_insert__mono,axiom,
    ! [C: set_VEBT_VEBT,D: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(C),D)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(insert_VEBT_VEBT(A2,C)),insert_VEBT_VEBT(A2,D)) ) ).

% insert_mono
tff(fact_1517_insert__mono,axiom,
    ! [C: set_real,D: set_real,A2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(C),D)
     => aa_set_real_o(ord_less_eq_set_real(insert_real(A2,C)),insert_real(A2,D)) ) ).

% insert_mono
tff(fact_1518_insert__mono,axiom,
    ! [C: set_o,D: set_o,A2: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(C),D)
     => aa_set_o_o(ord_less_eq_set_o(insert_o((A2),C)),insert_o((A2),D)) ) ).

% insert_mono
tff(fact_1519_insert__mono,axiom,
    ! [C: set_nat,D: set_nat,A2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(C),D)
     => aa_set_nat_o(ord_less_eq_set_nat(insert_nat(A2,C)),insert_nat(A2,D)) ) ).

% insert_mono
tff(fact_1520_psubset__imp__ex__mem,axiom,
    ! [A: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_set_real(A),B)
     => ? [B3: real] : aa_set_real_o(member_real(B3),minus_minus_set_real(B,A)) ) ).

% psubset_imp_ex_mem
tff(fact_1521_psubset__imp__ex__mem,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A),B)
     => ? [B3: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B3),minus_5127226145743854075T_VEBT(B,A)) ) ).

% psubset_imp_ex_mem
tff(fact_1522_psubset__imp__ex__mem,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
     => ? [B3: int] : aa_set_int_o(member_int(B3),minus_minus_set_int(B,A)) ) ).

% psubset_imp_ex_mem
tff(fact_1523_psubset__imp__ex__mem,axiom,
    ! [A: set_complex,B: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A),B)
     => ? [B3: complex] : aa_set_complex_o(member_complex(B3),minus_811609699411566653omplex(B,A)) ) ).

% psubset_imp_ex_mem
tff(fact_1524_psubset__imp__ex__mem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
     => ? [B3: nat] : aa_set_nat_o(member_nat(B3),minus_minus_set_nat(B,A)) ) ).

% psubset_imp_ex_mem
tff(fact_1525_DiffD2,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A,B))
     => ~ aa_set_real_o(member_real(C2),B) ) ).

% DiffD2
tff(fact_1526_DiffD2,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A,B))
     => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B) ) ).

% DiffD2
tff(fact_1527_DiffD2,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),minus_minus_set_int(A,B))
     => ~ aa_set_int_o(member_int(C2),B) ) ).

% DiffD2
tff(fact_1528_DiffD2,axiom,
    ! [C2: complex,A: set_complex,B: set_complex] :
      ( aa_set_complex_o(member_complex(C2),minus_811609699411566653omplex(A,B))
     => ~ aa_set_complex_o(member_complex(C2),B) ) ).

% DiffD2
tff(fact_1529_DiffD2,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),minus_minus_set_nat(A,B))
     => ~ aa_set_nat_o(member_nat(C2),B) ) ).

% DiffD2
tff(fact_1530_DiffD1,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A,B))
     => aa_set_real_o(member_real(C2),A) ) ).

% DiffD1
tff(fact_1531_DiffD1,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A,B))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A) ) ).

% DiffD1
tff(fact_1532_DiffD1,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),minus_minus_set_int(A,B))
     => aa_set_int_o(member_int(C2),A) ) ).

% DiffD1
tff(fact_1533_DiffD1,axiom,
    ! [C2: complex,A: set_complex,B: set_complex] :
      ( aa_set_complex_o(member_complex(C2),minus_811609699411566653omplex(A,B))
     => aa_set_complex_o(member_complex(C2),A) ) ).

% DiffD1
tff(fact_1534_DiffD1,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),minus_minus_set_nat(A,B))
     => aa_set_nat_o(member_nat(C2),A) ) ).

% DiffD1
tff(fact_1535_DiffE,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( aa_set_real_o(member_real(C2),minus_minus_set_real(A,B))
     => ~ ( aa_set_real_o(member_real(C2),A)
         => aa_set_real_o(member_real(C2),B) ) ) ).

% DiffE
tff(fact_1536_DiffE,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),minus_5127226145743854075T_VEBT(A,B))
     => ~ ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A)
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),B) ) ) ).

% DiffE
tff(fact_1537_DiffE,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( aa_set_int_o(member_int(C2),minus_minus_set_int(A,B))
     => ~ ( aa_set_int_o(member_int(C2),A)
         => aa_set_int_o(member_int(C2),B) ) ) ).

% DiffE
tff(fact_1538_DiffE,axiom,
    ! [C2: complex,A: set_complex,B: set_complex] :
      ( aa_set_complex_o(member_complex(C2),minus_811609699411566653omplex(A,B))
     => ~ ( aa_set_complex_o(member_complex(C2),A)
         => aa_set_complex_o(member_complex(C2),B) ) ) ).

% DiffE
tff(fact_1539_DiffE,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(member_nat(C2),minus_minus_set_nat(A,B))
     => ~ ( aa_set_nat_o(member_nat(C2),A)
         => aa_set_nat_o(member_nat(C2),B) ) ) ).

% DiffE
tff(fact_1540_double__diff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B),C)
       => ( minus_minus_set_nat(B,minus_minus_set_nat(C,A)) = A ) ) ) ).

% double_diff
tff(fact_1541_Diff__subset,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A,B)),A) ).

% Diff_subset
tff(fact_1542_Diff__mono,axiom,
    ! [A: set_nat,C: set_nat,D: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),C)
     => ( aa_set_nat_o(ord_less_eq_set_nat(D),B)
       => aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A,B)),minus_minus_set_nat(C,D)) ) ) ).

% Diff_mono
tff(fact_1543_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(P),Q)),R) = aa_assn_assn(times_times_assn(P),aa_assn_assn(times_times_assn(Q),R)) ).

% assn_times_assoc
tff(fact_1544_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : aa_assn_assn(times_times_assn(P),Q) = aa_assn_assn(times_times_assn(Q),P) ).

% assn_times_comm
tff(fact_1545_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_1546_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( entails(P,Q)
     => ( entails(Q,R)
       => entails(P,R) ) ) ).

% ent_trans
tff(fact_1547_ent__refl,axiom,
    ! [P: assn] : entails(P,P) ).

% ent_refl
tff(fact_1548_ent__iffI,axiom,
    ! [A: assn,B: assn] :
      ( entails(A,B)
     => ( entails(B,A)
       => ( A = B ) ) ) ).

% ent_iffI
tff(fact_1549_mult__right__cancel,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A2),C2) = aa_complex_complex(times_times_complex(B2),C2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_1550_mult__right__cancel,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_1551_mult__right__cancel,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A2),C2) = aa_rat_rat(times_times_rat(B2),C2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_1552_mult__right__cancel,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_1553_mult__right__cancel,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(times_times_int(A2),C2) = aa_int_int(times_times_int(B2),C2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_right_cancel
tff(fact_1554_mult__left__cancel,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(C2),A2) = aa_complex_complex(times_times_complex(C2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_1555_mult__left__cancel,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_1556_mult__left__cancel,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(C2),A2) = aa_rat_rat(times_times_rat(C2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_1557_mult__left__cancel,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_1558_mult__left__cancel,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( aa_int_int(times_times_int(C2),A2) = aa_int_int(times_times_int(C2),B2) )
      <=> ( A2 = B2 ) ) ) ).

% mult_left_cancel
tff(fact_1559_no__zero__divisors,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( aa_complex_complex(times_times_complex(A2),B2) != zero_zero_complex ) ) ) ).

% no_zero_divisors
tff(fact_1560_no__zero__divisors,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_1561_no__zero__divisors,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(times_times_rat(A2),B2) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_1562_no__zero__divisors,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_1563_no__zero__divisors,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( aa_int_int(times_times_int(A2),B2) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_1564_divisors__zero,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A2),B2) = zero_zero_complex )
     => ( ( A2 = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divisors_zero
tff(fact_1565_divisors__zero,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) = zero_zero_real )
     => ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_1566_divisors__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),B2) = zero_zero_rat )
     => ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_1567_divisors__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = zero_zero_nat )
     => ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_1568_divisors__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(times_times_int(A2),B2) = zero_zero_int )
     => ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_1569_mult__not__zero,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A2),B2) != zero_zero_complex )
     => ( ( A2 != zero_zero_complex )
        & ( B2 != zero_zero_complex ) ) ) ).

% mult_not_zero
tff(fact_1570_mult__not__zero,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) != zero_zero_real )
     => ( ( A2 != zero_zero_real )
        & ( B2 != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_1571_mult__not__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),B2) != zero_zero_rat )
     => ( ( A2 != zero_zero_rat )
        & ( B2 != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_1572_mult__not__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) != zero_zero_nat )
     => ( ( A2 != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_1573_mult__not__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(times_times_int(A2),B2) != zero_zero_int )
     => ( ( A2 != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_1574_num_Osize_I5_J,axiom,
    ! [X22: num] : aa_num_nat(size_size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(5)
tff(fact_1575_combine__common__factor,axiom,
    ! [A2: real,E: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),E)),C2) ).

% combine_common_factor
tff(fact_1576_combine__common__factor,axiom,
    ! [A2: rat,E: rat,B2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A2),B2)),E)),C2) ).

% combine_common_factor
tff(fact_1577_combine__common__factor,axiom,
    ! [A2: nat,E: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),E)),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),E)),C2) ).

% combine_common_factor
tff(fact_1578_combine__common__factor,axiom,
    ! [A2: int,E: int,B2: int,C2: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),E)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),E)),C2)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(plus_plus_int(A2),B2)),E)),C2) ).

% combine_common_factor
tff(fact_1579_distrib__right,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% distrib_right
tff(fact_1580_distrib__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ).

% distrib_right
tff(fact_1581_distrib__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ).

% distrib_right
tff(fact_1582_distrib__right,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A2),B2)),C2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2)) ).

% distrib_right
tff(fact_1583_distrib__left,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ).

% distrib_left
tff(fact_1584_distrib__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),B2)),aa_rat_rat(times_times_rat(A2),C2)) ).

% distrib_left
tff(fact_1585_distrib__left,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)) ).

% distrib_left
tff(fact_1586_distrib__left,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(times_times_int(A2),aa_int_int(plus_plus_int(B2),C2)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),B2)),aa_int_int(times_times_int(A2),C2)) ).

% distrib_left
tff(fact_1587_comm__semiring__class_Odistrib,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% comm_semiring_class.distrib
tff(fact_1588_comm__semiring__class_Odistrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ).

% comm_semiring_class.distrib
tff(fact_1589_comm__semiring__class_Odistrib,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ).

% comm_semiring_class.distrib
tff(fact_1590_comm__semiring__class_Odistrib,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A2),B2)),C2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2)) ).

% comm_semiring_class.distrib
tff(fact_1591_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_1592_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),B2)),aa_rat_rat(times_times_rat(A2),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_1593_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(times_times_int(A2),aa_int_int(plus_plus_int(B2),C2)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),B2)),aa_int_int(times_times_int(A2),C2)) ).

% ring_class.ring_distribs(1)
tff(fact_1594_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_1595_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_1596_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(times_times_int(aa_int_int(plus_plus_int(A2),B2)),C2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2)) ).

% ring_class.ring_distribs(2)
tff(fact_1597_right__diff__distrib_H,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ).

% right_diff_distrib'
tff(fact_1598_right__diff__distrib_H,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(minus_minus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A2),B2)),aa_rat_rat(times_times_rat(A2),C2)) ).

% right_diff_distrib'
tff(fact_1599_right__diff__distrib_H,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)) ).

% right_diff_distrib'
tff(fact_1600_right__diff__distrib_H,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(times_times_int(A2),aa_int_int(minus_minus_int(B2),C2)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A2),B2)),aa_int_int(times_times_int(A2),C2)) ).

% right_diff_distrib'
tff(fact_1601_left__diff__distrib_H,axiom,
    ! [B2: real,C2: real,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)),A2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)) ).

% left_diff_distrib'
tff(fact_1602_left__diff__distrib_H,axiom,
    ! [B2: rat,C2: rat,A2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B2),C2)),A2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(B2),A2)),aa_rat_rat(times_times_rat(C2),A2)) ).

% left_diff_distrib'
tff(fact_1603_left__diff__distrib_H,axiom,
    ! [B2: nat,C2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),C2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)) ).

% left_diff_distrib'
tff(fact_1604_left__diff__distrib_H,axiom,
    ! [B2: int,C2: int,A2: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(B2),C2)),A2) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(B2),A2)),aa_int_int(times_times_int(C2),A2)) ).

% left_diff_distrib'
tff(fact_1605_right__diff__distrib,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ).

% right_diff_distrib
tff(fact_1606_right__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(minus_minus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A2),B2)),aa_rat_rat(times_times_rat(A2),C2)) ).

% right_diff_distrib
tff(fact_1607_right__diff__distrib,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(times_times_int(A2),aa_int_int(minus_minus_int(B2),C2)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A2),B2)),aa_int_int(times_times_int(A2),C2)) ).

% right_diff_distrib
tff(fact_1608_left__diff__distrib,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% left_diff_distrib
tff(fact_1609_left__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A2),B2)),C2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ).

% left_diff_distrib
tff(fact_1610_left__diff__distrib,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(A2),B2)),C2) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2)) ).

% left_diff_distrib
tff(fact_1611_add__divide__distrib,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A2),B2)),C2) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A2),C2)),aa_complex_complex(divide1717551699836669952omplex(B2),C2)) ).

% add_divide_distrib
tff(fact_1612_add__divide__distrib,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) ).

% add_divide_distrib
tff(fact_1613_add__divide__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2)) ).

% add_divide_distrib
tff(fact_1614_divide__divide__eq__left_H,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A2),B2)),C2) = aa_complex_complex(divide1717551699836669952omplex(A2),aa_complex_complex(times_times_complex(C2),B2)) ).

% divide_divide_eq_left'
tff(fact_1615_divide__divide__eq__left_H,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ).

% divide_divide_eq_left'
tff(fact_1616_divide__divide__eq__left_H,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A2),B2)),C2) = aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(times_times_rat(C2),B2)) ).

% divide_divide_eq_left'
tff(fact_1617_divide__divide__times__eq,axiom,
    ! [Xc: complex,Ya: complex,Z: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(Xc),Ya)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xc),W)),aa_complex_complex(times_times_complex(Ya),Z)) ).

% divide_divide_times_eq
tff(fact_1618_divide__divide__times__eq,axiom,
    ! [Xc: real,Ya: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z)) ).

% divide_divide_times_eq
tff(fact_1619_divide__divide__times__eq,axiom,
    ! [Xc: rat,Ya: rat,Z: rat,W: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(Xc),Ya)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xc),W)),aa_rat_rat(times_times_rat(Ya),Z)) ).

% divide_divide_times_eq
tff(fact_1620_times__divide__times__eq,axiom,
    ! [Xc: complex,Ya: complex,Z: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(Xc),Ya)),aa_complex_complex(divide1717551699836669952omplex(Z),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xc),Z)),aa_complex_complex(times_times_complex(Ya),W)) ).

% times_divide_times_eq
tff(fact_1621_times__divide__times__eq,axiom,
    ! [Xc: real,Ya: real,Z: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Z),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),W)) ).

% times_divide_times_eq
tff(fact_1622_times__divide__times__eq,axiom,
    ! [Xc: rat,Ya: rat,Z: rat,W: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(divide_divide_rat(Xc),Ya)),aa_rat_rat(divide_divide_rat(Z),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(Xc),Z)),aa_rat_rat(times_times_rat(Ya),W)) ).

% times_divide_times_eq
tff(fact_1623_diff__divide__distrib,axiom,
    ! [A2: complex,B2: complex,C2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A2),B2)),C2) = aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A2),C2)),aa_complex_complex(divide1717551699836669952omplex(B2),C2)) ).

% diff_divide_distrib
tff(fact_1624_diff__divide__distrib,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) ).

% diff_divide_distrib
tff(fact_1625_diff__divide__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A2),B2)),C2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2)) ).

% diff_divide_distrib
tff(fact_1626_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),Nb) = A2 )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3)
                  & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y3),Nb) = A2 ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1627_realpow__pos__nth,axiom,
    ! [Nb: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => ? [R2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R2),Nb) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1628_singleton__inject,axiom,
    ! [A2: vEBT_VEBT,B2: vEBT_VEBT] :
      ( ( insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT) = insert_VEBT_VEBT(B2,bot_bo8194388402131092736T_VEBT) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_1629_singleton__inject,axiom,
    ! [A2: real,B2: real] :
      ( ( insert_real(A2,bot_bot_set_real) = insert_real(B2,bot_bot_set_real) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_1630_singleton__inject,axiom,
    ! [A2: $o,B2: $o] :
      ( ( insert_o((A2),bot_bot_set_o) = insert_o((B2),bot_bot_set_o) )
     => ( (A2)
      <=> (B2) ) ) ).

% singleton_inject
tff(fact_1631_singleton__inject,axiom,
    ! [A2: nat,B2: nat] :
      ( ( insert_nat(A2,bot_bot_set_nat) = insert_nat(B2,bot_bot_set_nat) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_1632_singleton__inject,axiom,
    ! [A2: int,B2: int] :
      ( ( insert_int(A2,bot_bot_set_int) = insert_int(B2,bot_bot_set_int) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_1633_insert__not__empty,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] : insert_VEBT_VEBT(A2,A) != bot_bo8194388402131092736T_VEBT ).

% insert_not_empty
tff(fact_1634_insert__not__empty,axiom,
    ! [A2: real,A: set_real] : insert_real(A2,A) != bot_bot_set_real ).

% insert_not_empty
tff(fact_1635_insert__not__empty,axiom,
    ! [A2: $o,A: set_o] : insert_o((A2),A) != bot_bot_set_o ).

% insert_not_empty
tff(fact_1636_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] : insert_nat(A2,A) != bot_bot_set_nat ).

% insert_not_empty
tff(fact_1637_insert__not__empty,axiom,
    ! [A2: int,A: set_int] : insert_int(A2,A) != bot_bot_set_int ).

% insert_not_empty
tff(fact_1638_doubleton__eq__iff,axiom,
    ! [A2: vEBT_VEBT,B2: vEBT_VEBT,C2: vEBT_VEBT,D2: vEBT_VEBT] :
      ( ( insert_VEBT_VEBT(A2,insert_VEBT_VEBT(B2,bot_bo8194388402131092736T_VEBT)) = insert_VEBT_VEBT(C2,insert_VEBT_VEBT(D2,bot_bo8194388402131092736T_VEBT)) )
    <=> ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_1639_doubleton__eq__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( insert_real(A2,insert_real(B2,bot_bot_set_real)) = insert_real(C2,insert_real(D2,bot_bot_set_real)) )
    <=> ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_1640_doubleton__eq__iff,axiom,
    ! [A2: $o,B2: $o,C2: $o,D2: $o] :
      ( ( insert_o((A2),insert_o((B2),bot_bot_set_o)) = insert_o((C2),insert_o((D2),bot_bot_set_o)) )
    <=> ( ( ( (A2)
          <=> (C2) )
          & ( (B2)
          <=> (D2) ) )
        | ( ( (A2)
          <=> (D2) )
          & ( (B2)
          <=> (C2) ) ) ) ) ).

% doubleton_eq_iff
tff(fact_1641_doubleton__eq__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( insert_nat(A2,insert_nat(B2,bot_bot_set_nat)) = insert_nat(C2,insert_nat(D2,bot_bot_set_nat)) )
    <=> ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_1642_doubleton__eq__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( insert_int(A2,insert_int(B2,bot_bot_set_int)) = insert_int(C2,insert_int(D2,bot_bot_set_int)) )
    <=> ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_1643_singleton__iff,axiom,
    ! [B2: vEBT_VEBT,A2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_1644_singleton__iff,axiom,
    ! [B2: complex,A2: complex] :
      ( aa_set_complex_o(member_complex(B2),insert_complex(A2,bot_bot_set_complex))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_1645_singleton__iff,axiom,
    ! [B2: real,A2: real] :
      ( aa_set_real_o(member_real(B2),insert_real(A2,bot_bot_set_real))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_1646_singleton__iff,axiom,
    ! [B2: $o,A2: $o] :
      ( aa_set_o_o(member_o((B2)),insert_o((A2),bot_bot_set_o))
    <=> ( (B2)
      <=> (A2) ) ) ).

% singleton_iff
tff(fact_1647_singleton__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_set_nat_o(member_nat(B2),insert_nat(A2,bot_bot_set_nat))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_1648_singleton__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_set_int_o(member_int(B2),insert_int(A2,bot_bot_set_int))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_1649_singletonD,axiom,
    ! [B2: vEBT_VEBT,A2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_1650_singletonD,axiom,
    ! [B2: complex,A2: complex] :
      ( aa_set_complex_o(member_complex(B2),insert_complex(A2,bot_bot_set_complex))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_1651_singletonD,axiom,
    ! [B2: real,A2: real] :
      ( aa_set_real_o(member_real(B2),insert_real(A2,bot_bot_set_real))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_1652_singletonD,axiom,
    ! [B2: $o,A2: $o] :
      ( aa_set_o_o(member_o((B2)),insert_o((A2),bot_bot_set_o))
     => ( (B2)
      <=> (A2) ) ) ).

% singletonD
tff(fact_1653_singletonD,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_set_nat_o(member_nat(B2),insert_nat(A2,bot_bot_set_nat))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_1654_singletonD,axiom,
    ! [B2: int,A2: int] :
      ( aa_set_int_o(member_int(B2),insert_int(A2,bot_bot_set_int))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_1655_subset__singleton__iff,axiom,
    ! [X: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(X),insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))
    <=> ( ( X = bot_bo8194388402131092736T_VEBT )
        | ( X = insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singleton_iff
tff(fact_1656_subset__singleton__iff,axiom,
    ! [X: set_real,A2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(X),insert_real(A2,bot_bot_set_real))
    <=> ( ( X = bot_bot_set_real )
        | ( X = insert_real(A2,bot_bot_set_real) ) ) ) ).

% subset_singleton_iff
tff(fact_1657_subset__singleton__iff,axiom,
    ! [X: set_o,A2: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(X),insert_o((A2),bot_bot_set_o))
    <=> ( ( X = bot_bot_set_o )
        | ( X = insert_o((A2),bot_bot_set_o) ) ) ) ).

% subset_singleton_iff
tff(fact_1658_subset__singleton__iff,axiom,
    ! [X: set_int,A2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(X),insert_int(A2,bot_bot_set_int))
    <=> ( ( X = bot_bot_set_int )
        | ( X = insert_int(A2,bot_bot_set_int) ) ) ) ).

% subset_singleton_iff
tff(fact_1659_subset__singleton__iff,axiom,
    ! [X: set_nat,A2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(X),insert_nat(A2,bot_bot_set_nat))
    <=> ( ( X = bot_bot_set_nat )
        | ( X = insert_nat(A2,bot_bot_set_nat) ) ) ) ).

% subset_singleton_iff
tff(fact_1660_subset__singletonD,axiom,
    ! [A: set_VEBT_VEBT,Xc: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(Xc,bot_bo8194388402131092736T_VEBT))
     => ( ( A = bot_bo8194388402131092736T_VEBT )
        | ( A = insert_VEBT_VEBT(Xc,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% subset_singletonD
tff(fact_1661_subset__singletonD,axiom,
    ! [A: set_real,Xc: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),insert_real(Xc,bot_bot_set_real))
     => ( ( A = bot_bot_set_real )
        | ( A = insert_real(Xc,bot_bot_set_real) ) ) ) ).

% subset_singletonD
tff(fact_1662_subset__singletonD,axiom,
    ! [A: set_o,Xc: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),insert_o((Xc),bot_bot_set_o))
     => ( ( A = bot_bot_set_o )
        | ( A = insert_o((Xc),bot_bot_set_o) ) ) ) ).

% subset_singletonD
tff(fact_1663_subset__singletonD,axiom,
    ! [A: set_int,Xc: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),insert_int(Xc,bot_bot_set_int))
     => ( ( A = bot_bot_set_int )
        | ( A = insert_int(Xc,bot_bot_set_int) ) ) ) ).

% subset_singletonD
tff(fact_1664_subset__singletonD,axiom,
    ! [A: set_nat,Xc: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(Xc,bot_bot_set_nat))
     => ( ( A = bot_bot_set_nat )
        | ( A = insert_nat(Xc,bot_bot_set_nat) ) ) ) ).

% subset_singletonD
tff(fact_1665_insert__Diff__if,axiom,
    ! [Xc: $o,A: set_o,B: set_o] :
      minus_minus_set_o(insert_o((Xc),A),B) = $ite(aa_set_o_o(member_o((Xc)),B),minus_minus_set_o(A,B),insert_o((Xc),minus_minus_set_o(A,B))) ).

% insert_Diff_if
tff(fact_1666_insert__Diff__if,axiom,
    ! [Xc: real,A: set_real,B: set_real] :
      minus_minus_set_real(insert_real(Xc,A),B) = $ite(aa_set_real_o(member_real(Xc),B),minus_minus_set_real(A,B),insert_real(Xc,minus_minus_set_real(A,B))) ).

% insert_Diff_if
tff(fact_1667_insert__Diff__if,axiom,
    ! [Xc: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xc,A),B) = $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),B),minus_5127226145743854075T_VEBT(A,B),insert_VEBT_VEBT(Xc,minus_5127226145743854075T_VEBT(A,B))) ).

% insert_Diff_if
tff(fact_1668_insert__Diff__if,axiom,
    ! [Xc: int,A: set_int,B: set_int] :
      minus_minus_set_int(insert_int(Xc,A),B) = $ite(aa_set_int_o(member_int(Xc),B),minus_minus_set_int(A,B),insert_int(Xc,minus_minus_set_int(A,B))) ).

% insert_Diff_if
tff(fact_1669_insert__Diff__if,axiom,
    ! [Xc: complex,A: set_complex,B: set_complex] :
      minus_811609699411566653omplex(insert_complex(Xc,A),B) = $ite(aa_set_complex_o(member_complex(Xc),B),minus_811609699411566653omplex(A,B),insert_complex(Xc,minus_811609699411566653omplex(A,B))) ).

% insert_Diff_if
tff(fact_1670_insert__Diff__if,axiom,
    ! [Xc: nat,A: set_nat,B: set_nat] :
      minus_minus_set_nat(insert_nat(Xc,A),B) = $ite(aa_set_nat_o(member_nat(Xc),B),minus_minus_set_nat(A,B),insert_nat(Xc,minus_minus_set_nat(A,B))) ).

% insert_Diff_if
tff(fact_1671_subset__Diff__insert,axiom,
    ! [A: set_o,B: set_o,Xc: $o,C: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),minus_minus_set_o(B,insert_o((Xc),C)))
    <=> ( aa_set_o_o(ord_less_eq_set_o(A),minus_minus_set_o(B,C))
        & ~ aa_set_o_o(member_o((Xc)),A) ) ) ).

% subset_Diff_insert
tff(fact_1672_subset__Diff__insert,axiom,
    ! [A: set_real,B: set_real,Xc: real,C: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),minus_minus_set_real(B,insert_real(Xc,C)))
    <=> ( aa_set_real_o(ord_less_eq_set_real(A),minus_minus_set_real(B,C))
        & ~ aa_set_real_o(member_real(Xc),A) ) ) ).

% subset_Diff_insert
tff(fact_1673_subset__Diff__insert,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,Xc: vEBT_VEBT,C: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),minus_5127226145743854075T_VEBT(B,insert_VEBT_VEBT(Xc,C)))
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),minus_5127226145743854075T_VEBT(B,C))
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),A) ) ) ).

% subset_Diff_insert
tff(fact_1674_subset__Diff__insert,axiom,
    ! [A: set_int,B: set_int,Xc: int,C: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),minus_minus_set_int(B,insert_int(Xc,C)))
    <=> ( aa_set_int_o(ord_less_eq_set_int(A),minus_minus_set_int(B,C))
        & ~ aa_set_int_o(member_int(Xc),A) ) ) ).

% subset_Diff_insert
tff(fact_1675_subset__Diff__insert,axiom,
    ! [A: set_complex,B: set_complex,Xc: complex,C: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),minus_811609699411566653omplex(B,insert_complex(Xc,C)))
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(A),minus_811609699411566653omplex(B,C))
        & ~ aa_set_complex_o(member_complex(Xc),A) ) ) ).

% subset_Diff_insert
tff(fact_1676_subset__Diff__insert,axiom,
    ! [A: set_nat,B: set_nat,Xc: nat,C: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),minus_minus_set_nat(B,insert_nat(Xc,C)))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A),minus_minus_set_nat(B,C))
        & ~ aa_set_nat_o(member_nat(Xc),A) ) ) ).

% subset_Diff_insert
tff(fact_1677_ent__star__mono,axiom,
    ! [P: assn,P2: assn,Q: assn,Q2: assn] :
      ( entails(P,P2)
     => ( entails(Q,Q2)
       => entails(aa_assn_assn(times_times_assn(P),Q),aa_assn_assn(times_times_assn(P2),Q2)) ) ) ).

% ent_star_mono
tff(fact_1678_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1679_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1680_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1681_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1682_zero__le__mult__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_1683_zero__le__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_1684_zero__le__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A2),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_1685_mult__nonneg__nonpos2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1686_mult__nonneg__nonpos2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(B2),A2)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1687_mult__nonneg__nonpos2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1688_mult__nonneg__nonpos2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(B2),A2)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1689_mult__nonpos__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_1690_mult__nonpos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_1691_mult__nonpos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_1692_mult__nonpos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),B2)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_1693_mult__nonneg__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_1694_mult__nonneg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_1695_mult__nonneg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_1696_mult__nonneg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),B2)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_1697_mult__nonneg__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1698_mult__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1699_mult__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1700_mult__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A2),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1701_split__mult__neg__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_1702_split__mult__neg__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_1703_split__mult__neg__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_1704_split__mult__neg__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),B2)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_1705_mult__le__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_1706_mult__le__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_1707_mult__le__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),B2)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_1708_mult__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ) ) ).

% mult_right_mono
tff(fact_1709_mult__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ) ) ).

% mult_right_mono
tff(fact_1710_mult__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ) ) ).

% mult_right_mono
tff(fact_1711_mult__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2)) ) ) ).

% mult_right_mono
tff(fact_1712_mult__right__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_1713_mult__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_1714_mult__right__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2)) ) ) ).

% mult_right_mono_neg
tff(fact_1715_mult__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% mult_left_mono
tff(fact_1716_mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)) ) ) ).

% mult_left_mono
tff(fact_1717_mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) ) ) ).

% mult_left_mono
tff(fact_1718_mult__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2)) ) ) ).

% mult_left_mono
tff(fact_1719_mult__nonpos__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1720_mult__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1721_mult__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A2),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1722_mult__left__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_1723_mult__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_1724_mult__left__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_1725_split__mult__pos__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) ) ).

% split_mult_pos_le
tff(fact_1726_split__mult__pos__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)) ) ).

% split_mult_pos_le
tff(fact_1727_split__mult__pos__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A2),B2)) ) ).

% split_mult_pos_le
tff(fact_1728_zero__le__square,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2)) ).

% zero_le_square
tff(fact_1729_zero__le__square,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),A2)) ).

% zero_le_square
tff(fact_1730_zero__le__square,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(times_times_int(A2),A2)) ).

% zero_le_square
tff(fact_1731_mult__mono_H,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_1732_mult__mono_H,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_1733_mult__mono_H,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_1734_mult__mono_H,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_1735_mult__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2)) ) ) ) ) ).

% mult_mono
tff(fact_1736_mult__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2)) ) ) ) ) ).

% mult_mono
tff(fact_1737_mult__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2)) ) ) ) ) ).

% mult_mono
tff(fact_1738_mult__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),D2)) ) ) ) ) ).

% mult_mono
tff(fact_1739_add__less__zeroD,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_1740_add__less__zeroD,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(Xc),Ya)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_1741_add__less__zeroD,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(Xc),Ya)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_1742_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1743_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1744_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1745_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1746_mult__less__cancel__right__disj,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1747_mult__less__cancel__right__disj,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1748_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1749_mult__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_1750_mult__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_1751_mult__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_1752_mult__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2)) ) ) ).

% mult_strict_right_mono
tff(fact_1753_mult__strict__right__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1754_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1755_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1756_mult__less__cancel__left__disj,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1757_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1758_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1759_mult__strict__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_1760_mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_1761_mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_1762_mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_1763_mult__strict__left__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1764_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1765_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1766_mult__less__cancel__left__pos,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1767_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1768_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1769_mult__less__cancel__left__neg,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1770_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1771_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1772_zero__less__mult__pos2,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_1773_zero__less__mult__pos2,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(B2),A2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_1774_zero__less__mult__pos2,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_1775_zero__less__mult__pos2,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(B2),A2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_1776_zero__less__mult__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ).

% zero_less_mult_pos
tff(fact_1777_zero__less__mult__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ).

% zero_less_mult_pos
tff(fact_1778_zero__less__mult__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ) ).

% zero_less_mult_pos
tff(fact_1779_zero__less__mult__pos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A2),B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ).

% zero_less_mult_pos
tff(fact_1780_zero__less__mult__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_1781_zero__less__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_1782_zero__less__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A2),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_1783_mult__pos__neg2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_1784_mult__pos__neg2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(B2),A2)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_1785_mult__pos__neg2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_1786_mult__pos__neg2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(B2),A2)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_1787_mult__pos__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) ) ) ).

% mult_pos_pos
tff(fact_1788_mult__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)) ) ) ).

% mult_pos_pos
tff(fact_1789_mult__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)) ) ) ).

% mult_pos_pos
tff(fact_1790_mult__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A2),B2)) ) ) ).

% mult_pos_pos
tff(fact_1791_mult__pos__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_1792_mult__pos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_1793_mult__pos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_1794_mult__pos__neg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),B2)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_1795_mult__neg__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_1796_mult__neg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_1797_mult__neg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_1798_mult__neg__pos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),B2)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_1799_mult__less__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_1800_mult__less__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_1801_mult__less__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),B2)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_1802_not__square__less__zero,axiom,
    ! [A2: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2)),zero_zero_real) ).

% not_square_less_zero
tff(fact_1803_not__square__less__zero,axiom,
    ! [A2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),A2)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_1804_not__square__less__zero,axiom,
    ! [A2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),A2)),zero_zero_int) ).

% not_square_less_zero
tff(fact_1805_mult__neg__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) ) ) ).

% mult_neg_neg
tff(fact_1806_mult__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2)) ) ) ).

% mult_neg_neg
tff(fact_1807_mult__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(times_times_int(A2),B2)) ) ) ).

% mult_neg_neg
tff(fact_1808_divide__right__mono__neg,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)) ) ) ).

% divide_right_mono_neg
tff(fact_1809_divide__right__mono__neg,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),C2)),aa_rat_rat(divide_divide_rat(A2),C2)) ) ) ).

% divide_right_mono_neg
tff(fact_1810_divide__nonpos__nonpos,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) ) ) ).

% divide_nonpos_nonpos
tff(fact_1811_divide__nonpos__nonpos,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xc),Ya)) ) ) ).

% divide_nonpos_nonpos
tff(fact_1812_divide__nonpos__nonneg,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_1813_divide__nonpos__nonneg,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xc),Ya)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_1814_divide__nonneg__nonpos,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_1815_divide__nonneg__nonpos,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xc),Ya)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_1816_divide__nonneg__nonneg,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) ) ) ).

% divide_nonneg_nonneg
tff(fact_1817_divide__nonneg__nonneg,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xc),Ya)) ) ) ).

% divide_nonneg_nonneg
tff(fact_1818_zero__le__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_1819_zero__le__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A2),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_1820_divide__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) ) ) ).

% divide_right_mono
tff(fact_1821_divide__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2)) ) ) ).

% divide_right_mono
tff(fact_1822_divide__le__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ) ) ).

% divide_le_0_iff
tff(fact_1823_divide__le__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A2),B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ) ) ).

% divide_le_0_iff
tff(fact_1824_divide__strict__right__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1825_divide__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1826_divide__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) ) ) ).

% divide_strict_right_mono
tff(fact_1827_divide__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2)) ) ) ).

% divide_strict_right_mono
tff(fact_1828_zero__less__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_1829_zero__less__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A2),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_1830_divide__less__cancel,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) )
        & ( C2 != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_1831_divide__less__cancel,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) )
        & ( C2 != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_1832_divide__less__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)),zero_zero_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ) ) ).

% divide_less_0_iff
tff(fact_1833_divide__less__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A2),B2)),zero_zero_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ) ) ).

% divide_less_0_iff
tff(fact_1834_divide__pos__pos,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) ) ) ).

% divide_pos_pos
tff(fact_1835_divide__pos__pos,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xc),Ya)) ) ) ).

% divide_pos_pos
tff(fact_1836_divide__pos__neg,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_1837_divide__pos__neg,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xc),Ya)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_1838_divide__neg__pos,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_1839_divide__neg__pos,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xc),Ya)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_1840_divide__neg__neg,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) ) ) ).

% divide_neg_neg
tff(fact_1841_divide__neg__neg,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xc),Ya)) ) ) ).

% divide_neg_neg
tff(fact_1842_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,Nb: real,J2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),K))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Nb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Nb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),K))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K)),J2) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1843_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,Nb: rat,J2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J2),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(plus_plus_rat(J2),K))
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Nb),K)),J2) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1844_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,Nb: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Nb)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),K))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),J2) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1845_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,Nb: int,J2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(plus_plus_int(J2),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(plus_plus_int(J2),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(Nb),K)),J2) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1846_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_1847_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_1848_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_1849_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(minus_minus_int(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_1850_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: real,B2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1851_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_rat(plus_plus_rat(B2),aa_rat_rat(minus_minus_rat(A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1852_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1853_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_int(plus_plus_int(B2),aa_int_int(minus_minus_int(A2),B2)) = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1854_nonzero__eq__divide__eq,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( A2 = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
      <=> ( aa_complex_complex(times_times_complex(A2),C2) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1855_nonzero__eq__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1856_nonzero__eq__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( A2 = aa_rat_rat(divide_divide_rat(B2),C2) )
      <=> ( aa_rat_rat(times_times_rat(A2),C2) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1857_nonzero__divide__eq__eq,axiom,
    ! [C2: complex,B2: complex,A2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C2) = A2 )
      <=> ( B2 = aa_complex_complex(times_times_complex(A2),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1858_nonzero__divide__eq__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = A2 )
      <=> ( B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1859_nonzero__divide__eq__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B2),C2) = A2 )
      <=> ( B2 = aa_rat_rat(times_times_rat(A2),C2) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1860_eq__divide__imp,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A2),C2) = B2 )
       => ( A2 = aa_complex_complex(divide1717551699836669952omplex(B2),C2) ) ) ) ).

% eq_divide_imp
tff(fact_1861_eq__divide__imp,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = B2 )
       => ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) ) ) ) ).

% eq_divide_imp
tff(fact_1862_eq__divide__imp,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( aa_rat_rat(times_times_rat(A2),C2) = B2 )
       => ( A2 = aa_rat_rat(divide_divide_rat(B2),C2) ) ) ) ).

% eq_divide_imp
tff(fact_1863_divide__eq__imp,axiom,
    ! [C2: complex,B2: complex,A2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( B2 = aa_complex_complex(times_times_complex(A2),C2) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B2),C2) = A2 ) ) ) ).

% divide_eq_imp
tff(fact_1864_divide__eq__imp,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = A2 ) ) ) ).

% divide_eq_imp
tff(fact_1865_divide__eq__imp,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( B2 = aa_rat_rat(times_times_rat(A2),C2) )
       => ( aa_rat_rat(divide_divide_rat(B2),C2) = A2 ) ) ) ).

% divide_eq_imp
tff(fact_1866_eq__divide__eq,axiom,
    ! [A2: complex,B2: complex,C2: complex] :
      ( ( A2 = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(A2),C2) = B2,A2 = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_1867_eq__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
    <=> $ite(C2 != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = B2,A2 = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_1868_eq__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_rat(divide_divide_rat(B2),C2) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(times_times_rat(A2),C2) = B2,A2 = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_1869_divide__eq__eq,axiom,
    ! [B2: complex,C2: complex,A2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C2) = A2 )
    <=> $ite(C2 != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(A2),C2),A2 = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_1870_divide__eq__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = A2 )
    <=> $ite(C2 != zero_zero_real,B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2),A2 = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_1871_divide__eq__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B2),C2) = A2 )
    <=> $ite(C2 != zero_zero_rat,B2 = aa_rat_rat(times_times_rat(A2),C2),A2 = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_1872_frac__eq__eq,axiom,
    ! [Ya: complex,Z: complex,Xc: complex,W: complex] :
      ( ( Ya != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(Xc),Ya) = aa_complex_complex(divide1717551699836669952omplex(W),Z) )
        <=> ( aa_complex_complex(times_times_complex(Xc),Z) = aa_complex_complex(times_times_complex(W),Ya) ) ) ) ) ).

% frac_eq_eq
tff(fact_1873_frac__eq__eq,axiom,
    ! [Ya: real,Z: real,Xc: real,W: real] :
      ( ( Ya != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Z) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Ya) ) ) ) ) ).

% frac_eq_eq
tff(fact_1874_frac__eq__eq,axiom,
    ! [Ya: rat,Z: rat,Xc: rat,W: rat] :
      ( ( Ya != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(Xc),Ya) = aa_rat_rat(divide_divide_rat(W),Z) )
        <=> ( aa_rat_rat(times_times_rat(Xc),Z) = aa_rat_rat(times_times_rat(W),Ya) ) ) ) ) ).

% frac_eq_eq
tff(fact_1875_four__x__squared,axiom,
    ! [Xc: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_1876_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_1877_square__diff__square__factored,axiom,
    ! [Xc: real,Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Ya)) ).

% square_diff_square_factored
tff(fact_1878_square__diff__square__factored,axiom,
    ! [Xc: rat,Ya: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xc),Xc)),aa_rat_rat(times_times_rat(Ya),Ya)) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xc),Ya)),aa_rat_rat(minus_minus_rat(Xc),Ya)) ).

% square_diff_square_factored
tff(fact_1879_square__diff__square__factored,axiom,
    ! [Xc: int,Ya: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(Xc),Xc)),aa_int_int(times_times_int(Ya),Ya)) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(Xc),Ya)),aa_int_int(minus_minus_int(Xc),Ya)) ).

% square_diff_square_factored
tff(fact_1880_eq__add__iff2,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2) )
    <=> ( C2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_1881_eq__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2) )
    <=> ( C2 = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B2),A2)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_1882_eq__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),E)),C2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),E)),D2) )
    <=> ( C2 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B2),A2)),E)),D2) ) ) ).

% eq_add_iff2
tff(fact_1883_eq__add__iff1,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),E)),C2) = D2 ) ) ).

% eq_add_iff1
tff(fact_1884_eq__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2) )
    <=> ( aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A2),B2)),E)),C2) = D2 ) ) ).

% eq_add_iff1
tff(fact_1885_eq__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),E)),C2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),E)),D2) )
    <=> ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A2),B2)),E)),C2) = D2 ) ) ).

% eq_add_iff1
tff(fact_1886_div__mult__le,axiom,
    ! [A2: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),A2) ).

% div_mult_le
tff(fact_1887_Diff__insert__absorb,axiom,
    ! [Xc: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),A)
     => ( minus_5127226145743854075T_VEBT(insert_VEBT_VEBT(Xc,A),insert_VEBT_VEBT(Xc,bot_bo8194388402131092736T_VEBT)) = A ) ) ).

% Diff_insert_absorb
tff(fact_1888_Diff__insert__absorb,axiom,
    ! [Xc: complex,A: set_complex] :
      ( ~ aa_set_complex_o(member_complex(Xc),A)
     => ( minus_811609699411566653omplex(insert_complex(Xc,A),insert_complex(Xc,bot_bot_set_complex)) = A ) ) ).

% Diff_insert_absorb
tff(fact_1889_Diff__insert__absorb,axiom,
    ! [Xc: real,A: set_real] :
      ( ~ aa_set_real_o(member_real(Xc),A)
     => ( minus_minus_set_real(insert_real(Xc,A),insert_real(Xc,bot_bot_set_real)) = A ) ) ).

% Diff_insert_absorb
tff(fact_1890_Diff__insert__absorb,axiom,
    ! [Xc: $o,A: set_o] :
      ( ~ aa_set_o_o(member_o((Xc)),A)
     => ( minus_minus_set_o(insert_o((Xc),A),insert_o((Xc),bot_bot_set_o)) = A ) ) ).

% Diff_insert_absorb
tff(fact_1891_Diff__insert__absorb,axiom,
    ! [Xc: int,A: set_int] :
      ( ~ aa_set_int_o(member_int(Xc),A)
     => ( minus_minus_set_int(insert_int(Xc,A),insert_int(Xc,bot_bot_set_int)) = A ) ) ).

% Diff_insert_absorb
tff(fact_1892_Diff__insert__absorb,axiom,
    ! [Xc: nat,A: set_nat] :
      ( ~ aa_set_nat_o(member_nat(Xc),A)
     => ( minus_minus_set_nat(insert_nat(Xc,A),insert_nat(Xc,bot_bot_set_nat)) = A ) ) ).

% Diff_insert_absorb
tff(fact_1893_Diff__insert2,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT,B: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,B)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)),B) ).

% Diff_insert2
tff(fact_1894_Diff__insert2,axiom,
    ! [A: set_real,A2: real,B: set_real] : minus_minus_set_real(A,insert_real(A2,B)) = minus_minus_set_real(minus_minus_set_real(A,insert_real(A2,bot_bot_set_real)),B) ).

% Diff_insert2
tff(fact_1895_Diff__insert2,axiom,
    ! [A: set_o,A2: $o,B: set_o] : minus_minus_set_o(A,insert_o((A2),B)) = minus_minus_set_o(minus_minus_set_o(A,insert_o((A2),bot_bot_set_o)),B) ).

% Diff_insert2
tff(fact_1896_Diff__insert2,axiom,
    ! [A: set_int,A2: int,B: set_int] : minus_minus_set_int(A,insert_int(A2,B)) = minus_minus_set_int(minus_minus_set_int(A,insert_int(A2,bot_bot_set_int)),B) ).

% Diff_insert2
tff(fact_1897_Diff__insert2,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] : minus_minus_set_nat(A,insert_nat(A2,B)) = minus_minus_set_nat(minus_minus_set_nat(A,insert_nat(A2,bot_bot_set_nat)),B) ).

% Diff_insert2
tff(fact_1898_insert__Diff,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A2),A)
     => ( insert_VEBT_VEBT(A2,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) = A ) ) ).

% insert_Diff
tff(fact_1899_insert__Diff,axiom,
    ! [A2: complex,A: set_complex] :
      ( aa_set_complex_o(member_complex(A2),A)
     => ( insert_complex(A2,minus_811609699411566653omplex(A,insert_complex(A2,bot_bot_set_complex))) = A ) ) ).

% insert_Diff
tff(fact_1900_insert__Diff,axiom,
    ! [A2: real,A: set_real] :
      ( aa_set_real_o(member_real(A2),A)
     => ( insert_real(A2,minus_minus_set_real(A,insert_real(A2,bot_bot_set_real))) = A ) ) ).

% insert_Diff
tff(fact_1901_insert__Diff,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(member_o((A2)),A)
     => ( insert_o((A2),minus_minus_set_o(A,insert_o((A2),bot_bot_set_o))) = A ) ) ).

% insert_Diff
tff(fact_1902_insert__Diff,axiom,
    ! [A2: int,A: set_int] :
      ( aa_set_int_o(member_int(A2),A)
     => ( insert_int(A2,minus_minus_set_int(A,insert_int(A2,bot_bot_set_int))) = A ) ) ).

% insert_Diff
tff(fact_1903_insert__Diff,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(A2),A)
     => ( insert_nat(A2,minus_minus_set_nat(A,insert_nat(A2,bot_bot_set_nat))) = A ) ) ).

% insert_Diff
tff(fact_1904_Diff__insert,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT,B: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,B)) = minus_5127226145743854075T_VEBT(minus_5127226145743854075T_VEBT(A,B),insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT)) ).

% Diff_insert
tff(fact_1905_Diff__insert,axiom,
    ! [A: set_real,A2: real,B: set_real] : minus_minus_set_real(A,insert_real(A2,B)) = minus_minus_set_real(minus_minus_set_real(A,B),insert_real(A2,bot_bot_set_real)) ).

% Diff_insert
tff(fact_1906_Diff__insert,axiom,
    ! [A: set_o,A2: $o,B: set_o] : minus_minus_set_o(A,insert_o((A2),B)) = minus_minus_set_o(minus_minus_set_o(A,B),insert_o((A2),bot_bot_set_o)) ).

% Diff_insert
tff(fact_1907_Diff__insert,axiom,
    ! [A: set_int,A2: int,B: set_int] : minus_minus_set_int(A,insert_int(A2,B)) = minus_minus_set_int(minus_minus_set_int(A,B),insert_int(A2,bot_bot_set_int)) ).

% Diff_insert
tff(fact_1908_Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] : minus_minus_set_nat(A,insert_nat(A2,B)) = minus_minus_set_nat(minus_minus_set_nat(A,B),insert_nat(A2,bot_bot_set_nat)) ).

% Diff_insert
tff(fact_1909_psubset__insert__iff,axiom,
    ! [A: set_VEBT_VEBT,Xc: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A),insert_VEBT_VEBT(Xc,B))
    <=> $ite(
          aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),B),
          aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A),B),
          $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),A),aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xc,bot_bo8194388402131092736T_VEBT))),B),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B)) ) ) ).

% psubset_insert_iff
tff(fact_1910_psubset__insert__iff,axiom,
    ! [A: set_complex,Xc: complex,B: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A),insert_complex(Xc,B))
    <=> $ite(
          aa_set_complex_o(member_complex(Xc),B),
          aa_set_complex_o(ord_less_set_complex(A),B),
          $ite(aa_set_complex_o(member_complex(Xc),A),aa_set_complex_o(ord_less_set_complex(minus_811609699411566653omplex(A,insert_complex(Xc,bot_bot_set_complex))),B),aa_set_complex_o(ord_le211207098394363844omplex(A),B)) ) ) ).

% psubset_insert_iff
tff(fact_1911_psubset__insert__iff,axiom,
    ! [A: set_real,Xc: real,B: set_real] :
      ( aa_set_real_o(ord_less_set_real(A),insert_real(Xc,B))
    <=> $ite(
          aa_set_real_o(member_real(Xc),B),
          aa_set_real_o(ord_less_set_real(A),B),
          $ite(aa_set_real_o(member_real(Xc),A),aa_set_real_o(ord_less_set_real(minus_minus_set_real(A,insert_real(Xc,bot_bot_set_real))),B),aa_set_real_o(ord_less_eq_set_real(A),B)) ) ) ).

% psubset_insert_iff
tff(fact_1912_psubset__insert__iff,axiom,
    ! [A: set_o,Xc: $o,B: set_o] :
      ( aa_set_o_o(ord_less_set_o(A),insert_o((Xc),B))
    <=> $ite(
          aa_set_o_o(member_o((Xc)),B),
          aa_set_o_o(ord_less_set_o(A),B),
          $ite(aa_set_o_o(member_o((Xc)),A),aa_set_o_o(ord_less_set_o(minus_minus_set_o(A,insert_o((Xc),bot_bot_set_o))),B),aa_set_o_o(ord_less_eq_set_o(A),B)) ) ) ).

% psubset_insert_iff
tff(fact_1913_psubset__insert__iff,axiom,
    ! [A: set_int,Xc: int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),insert_int(Xc,B))
    <=> $ite(
          aa_set_int_o(member_int(Xc),B),
          aa_set_int_o(ord_less_set_int(A),B),
          $ite(aa_set_int_o(member_int(Xc),A),aa_set_int_o(ord_less_set_int(minus_minus_set_int(A,insert_int(Xc,bot_bot_set_int))),B),aa_set_int_o(ord_less_eq_set_int(A),B)) ) ) ).

% psubset_insert_iff
tff(fact_1914_psubset__insert__iff,axiom,
    ! [A: set_nat,Xc: nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),insert_nat(Xc,B))
    <=> $ite(
          aa_set_nat_o(member_nat(Xc),B),
          aa_set_nat_o(ord_less_set_nat(A),B),
          $ite(aa_set_nat_o(member_nat(Xc),A),aa_set_nat_o(ord_less_set_nat(minus_minus_set_nat(A,insert_nat(Xc,bot_bot_set_nat))),B),aa_set_nat_o(ord_less_eq_set_nat(A),B)) ) ) ).

% psubset_insert_iff
tff(fact_1915_Diff__single__insert,axiom,
    ! [A: set_VEBT_VEBT,Xc: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xc,bot_bo8194388402131092736T_VEBT))),B)
     => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(Xc,B)) ) ).

% Diff_single_insert
tff(fact_1916_Diff__single__insert,axiom,
    ! [A: set_real,Xc: real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A,insert_real(Xc,bot_bot_set_real))),B)
     => aa_set_real_o(ord_less_eq_set_real(A),insert_real(Xc,B)) ) ).

% Diff_single_insert
tff(fact_1917_Diff__single__insert,axiom,
    ! [A: set_o,Xc: $o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A,insert_o((Xc),bot_bot_set_o))),B)
     => aa_set_o_o(ord_less_eq_set_o(A),insert_o((Xc),B)) ) ).

% Diff_single_insert
tff(fact_1918_Diff__single__insert,axiom,
    ! [A: set_int,Xc: int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A,insert_int(Xc,bot_bot_set_int))),B)
     => aa_set_int_o(ord_less_eq_set_int(A),insert_int(Xc,B)) ) ).

% Diff_single_insert
tff(fact_1919_Diff__single__insert,axiom,
    ! [A: set_nat,Xc: nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A,insert_nat(Xc,bot_bot_set_nat))),B)
     => aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(Xc,B)) ) ).

% Diff_single_insert
tff(fact_1920_subset__insert__iff,axiom,
    ! [A: set_VEBT_VEBT,Xc: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),insert_VEBT_VEBT(Xc,B))
    <=> $ite(aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),A),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(Xc,bot_bo8194388402131092736T_VEBT))),B),aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B)) ) ).

% subset_insert_iff
tff(fact_1921_subset__insert__iff,axiom,
    ! [A: set_complex,Xc: complex,B: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),insert_complex(Xc,B))
    <=> $ite(aa_set_complex_o(member_complex(Xc),A),aa_set_complex_o(ord_le211207098394363844omplex(minus_811609699411566653omplex(A,insert_complex(Xc,bot_bot_set_complex))),B),aa_set_complex_o(ord_le211207098394363844omplex(A),B)) ) ).

% subset_insert_iff
tff(fact_1922_subset__insert__iff,axiom,
    ! [A: set_real,Xc: real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),insert_real(Xc,B))
    <=> $ite(aa_set_real_o(member_real(Xc),A),aa_set_real_o(ord_less_eq_set_real(minus_minus_set_real(A,insert_real(Xc,bot_bot_set_real))),B),aa_set_real_o(ord_less_eq_set_real(A),B)) ) ).

% subset_insert_iff
tff(fact_1923_subset__insert__iff,axiom,
    ! [A: set_o,Xc: $o,B: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),insert_o((Xc),B))
    <=> $ite(aa_set_o_o(member_o((Xc)),A),aa_set_o_o(ord_less_eq_set_o(minus_minus_set_o(A,insert_o((Xc),bot_bot_set_o))),B),aa_set_o_o(ord_less_eq_set_o(A),B)) ) ).

% subset_insert_iff
tff(fact_1924_subset__insert__iff,axiom,
    ! [A: set_int,Xc: int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),insert_int(Xc,B))
    <=> $ite(aa_set_int_o(member_int(Xc),A),aa_set_int_o(ord_less_eq_set_int(minus_minus_set_int(A,insert_int(Xc,bot_bot_set_int))),B),aa_set_int_o(ord_less_eq_set_int(A),B)) ) ).

% subset_insert_iff
tff(fact_1925_subset__insert__iff,axiom,
    ! [A: set_nat,Xc: nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),insert_nat(Xc,B))
    <=> $ite(aa_set_nat_o(member_nat(Xc),A),aa_set_nat_o(ord_less_eq_set_nat(minus_minus_set_nat(A,insert_nat(Xc,bot_bot_set_nat))),B),aa_set_nat_o(ord_less_eq_set_nat(A),B)) ) ).

% subset_insert_iff
tff(fact_1926_field__le__epsilon,axiom,
    ! [Xc: real,Ya: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ya),E2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ).

% field_le_epsilon
tff(fact_1927_field__le__epsilon,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_rat_rat(plus_plus_rat(Ya),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya) ) ).

% field_le_epsilon
tff(fact_1928_mult__less__le__imp__less,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1929_mult__less__le__imp__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1930_mult__less__le__imp__less,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1931_mult__less__le__imp__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1932_mult__le__less__imp__less,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1933_mult__le__less__imp__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1934_mult__le__less__imp__less,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1935_mult__le__less__imp__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1936_mult__right__le__imp__le,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_1937_mult__right__le__imp__le,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_1938_mult__right__le__imp__le,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_1939_mult__right__le__imp__le,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_1940_mult__left__le__imp__le,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_1941_mult__left__le__imp__le,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_1942_mult__left__le__imp__le,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_1943_mult__left__le__imp__le,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_1944_mult__le__cancel__left__pos,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1945_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1946_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1947_mult__le__cancel__left__neg,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1948_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1949_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1950_mult__less__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ) ).

% mult_less_cancel_right
tff(fact_1951_mult__less__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ) ).

% mult_less_cancel_right
tff(fact_1952_mult__less__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ) ) ).

% mult_less_cancel_right
tff(fact_1953_mult__strict__mono_H,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1954_mult__strict__mono_H,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1955_mult__strict__mono_H,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1956_mult__strict__mono_H,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1957_mult__right__less__imp__less,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_1958_mult__right__less__imp__less,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_1959_mult__right__less__imp__less,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_1960_mult__right__less__imp__less,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_1961_mult__less__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ) ).

% mult_less_cancel_left
tff(fact_1962_mult__less__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ) ).

% mult_less_cancel_left
tff(fact_1963_mult__less__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ) ) ).

% mult_less_cancel_left
tff(fact_1964_mult__strict__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_1965_mult__strict__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_1966_mult__strict__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_1967_mult__strict__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_1968_mult__left__less__imp__less,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_1969_mult__left__less__imp__less,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_1970_mult__left__less__imp__less,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_1971_mult__left__less__imp__less,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_1972_mult__le__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ) ).

% mult_le_cancel_right
tff(fact_1973_mult__le__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ) ).

% mult_le_cancel_right
tff(fact_1974_mult__le__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ) ) ).

% mult_le_cancel_right
tff(fact_1975_mult__le__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ) ).

% mult_le_cancel_left
tff(fact_1976_mult__le__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ) ).

% mult_le_cancel_left
tff(fact_1977_mult__le__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ) ) ).

% mult_le_cancel_left
tff(fact_1978_sum__squares__ge__zero,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya))) ).

% sum_squares_ge_zero
tff(fact_1979_sum__squares__ge__zero,axiom,
    ! [Xc: rat,Ya: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xc),Xc)),aa_rat_rat(times_times_rat(Ya),Ya))) ).

% sum_squares_ge_zero
tff(fact_1980_sum__squares__ge__zero,axiom,
    ! [Xc: int,Ya: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xc),Xc)),aa_int_int(times_times_int(Ya),Ya))) ).

% sum_squares_ge_zero
tff(fact_1981_divide__nonpos__pos,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_1982_divide__nonpos__pos,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xc),Ya)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_1983_divide__nonpos__neg,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) ) ) ).

% divide_nonpos_neg
tff(fact_1984_divide__nonpos__neg,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xc),Ya)) ) ) ).

% divide_nonpos_neg
tff(fact_1985_divide__nonneg__pos,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) ) ) ).

% divide_nonneg_pos
tff(fact_1986_divide__nonneg__pos,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xc),Ya)) ) ) ).

% divide_nonneg_pos
tff(fact_1987_divide__nonneg__neg,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_1988_divide__nonneg__neg,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xc),Ya)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_1989_divide__le__cancel,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ) ).

% divide_le_cancel
tff(fact_1990_divide__le__cancel,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A2),C2)),aa_rat_rat(divide_divide_rat(B2),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ) ).

% divide_le_cancel
tff(fact_1991_frac__less2,axiom,
    ! [Xc: real,Ya: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ya),W)) ) ) ) ) ).

% frac_less2
tff(fact_1992_frac__less2,axiom,
    ! [Xc: rat,Ya: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xc),Z)),aa_rat_rat(divide_divide_rat(Ya),W)) ) ) ) ) ).

% frac_less2
tff(fact_1993_frac__less,axiom,
    ! [Xc: real,Ya: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ya),W)) ) ) ) ) ).

% frac_less
tff(fact_1994_frac__less,axiom,
    ! [Xc: rat,Ya: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xc),Z)),aa_rat_rat(divide_divide_rat(Ya),W)) ) ) ) ) ).

% frac_less
tff(fact_1995_frac__le,axiom,
    ! [Ya: real,Xc: real,W: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Z)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ya),W)) ) ) ) ) ).

% frac_le
tff(fact_1996_frac__le,axiom,
    ! [Ya: rat,Xc: rat,W: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Z)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xc),Z)),aa_rat_rat(divide_divide_rat(Ya),W)) ) ) ) ) ).

% frac_le
tff(fact_1997_not__sum__squares__lt__zero,axiom,
    ! [Xc: real,Ya: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_1998_not__sum__squares__lt__zero,axiom,
    ! [Xc: rat,Ya: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xc),Xc)),aa_rat_rat(times_times_rat(Ya),Ya))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_1999_not__sum__squares__lt__zero,axiom,
    ! [Xc: int,Ya: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xc),Xc)),aa_int_int(times_times_int(Ya),Ya))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_2000_divide__strict__left__mono__neg,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2001_divide__strict__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C2),A2)),aa_rat_rat(divide_divide_rat(C2),B2)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2002_divide__strict__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2003_divide__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(C2),A2)),aa_rat_rat(divide_divide_rat(C2),B2)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2004_mult__imp__less__div__pos,axiom,
    ! [Ya: real,Z: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Ya)),Xc)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2005_mult__imp__less__div__pos,axiom,
    ! [Ya: rat,Z: rat,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Z),Ya)),Xc)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(divide_divide_rat(Xc),Ya)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2006_mult__imp__div__pos__less,axiom,
    ! [Ya: real,Xc: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Ya))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2007_mult__imp__div__pos__less,axiom,
    ! [Ya: rat,Xc: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),aa_rat_rat(times_times_rat(Z),Ya))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xc),Ya)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2008_pos__less__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2) ) ) ).

% pos_less_divide_eq
tff(fact_2009_pos__less__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),B2) ) ) ).

% pos_less_divide_eq
tff(fact_2010_pos__divide__less__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ) ) ).

% pos_divide_less_eq
tff(fact_2011_pos__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C2)) ) ) ).

% pos_divide_less_eq
tff(fact_2012_neg__less__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ) ) ).

% neg_less_divide_eq
tff(fact_2013_neg__less__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C2)) ) ) ).

% neg_less_divide_eq
tff(fact_2014_neg__divide__less__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2) ) ) ).

% neg_divide_less_eq
tff(fact_2015_neg__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),B2) ) ) ).

% neg_divide_less_eq
tff(fact_2016_less__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)) ) ) ).

% less_divide_eq
tff(fact_2017_less__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_2018_divide__less__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)) ) ) ).

% divide_less_eq
tff(fact_2019_divide__less__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)) ) ) ).

% divide_less_eq
tff(fact_2020_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),E)),D2)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2021_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B2),A2)),E)),D2)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2022_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),E)),C2)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),E)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B2),A2)),E)),D2)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_2023_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),E)),C2)),D2) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2024_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A2),B2)),E)),C2)),D2) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2025_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),E)),C2)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),E)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A2),B2)),E)),C2)),D2) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_2026_divide__add__eq__iff,axiom,
    ! [Z: complex,Xc: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xc),Z)),Ya) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xc),aa_complex_complex(times_times_complex(Ya),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2027_divide__add__eq__iff,axiom,
    ! [Z: real,Xc: real,Ya: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Z)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2028_divide__add__eq__iff,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xc),Z)),Ya) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xc),aa_rat_rat(times_times_rat(Ya),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2029_add__divide__eq__iff,axiom,
    ! [Z: complex,Xc: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Xc),aa_complex_complex(divide1717551699836669952omplex(Ya),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xc),Z)),Ya)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2030_add__divide__eq__iff,axiom,
    ! [Z: real,Xc: real,Ya: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ya),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Z)),Ya)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2031_add__divide__eq__iff,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Xc),aa_rat_rat(divide_divide_rat(Ya),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xc),Z)),Ya)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2032_add__num__frac,axiom,
    ! [Ya: complex,Z: complex,Xc: complex] :
      ( ( Ya != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(divide1717551699836669952omplex(Xc),Ya)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xc),aa_complex_complex(times_times_complex(Z),Ya))),Ya) ) ) ).

% add_num_frac
tff(fact_2033_add__num__frac,axiom,
    ! [Ya: real,Z: real,Xc: real] :
      ( ( Ya != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Ya))),Ya) ) ) ).

% add_num_frac
tff(fact_2034_add__num__frac,axiom,
    ! [Ya: rat,Z: rat,Xc: rat] :
      ( ( Ya != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(Z),aa_rat_rat(divide_divide_rat(Xc),Ya)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xc),aa_rat_rat(times_times_rat(Z),Ya))),Ya) ) ) ).

% add_num_frac
tff(fact_2035_add__frac__num,axiom,
    ! [Ya: complex,Xc: complex,Z: complex] :
      ( ( Ya != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xc),Ya)),Z) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xc),aa_complex_complex(times_times_complex(Z),Ya))),Ya) ) ) ).

% add_frac_num
tff(fact_2036_add__frac__num,axiom,
    ! [Ya: real,Xc: real,Z: real] :
      ( ( Ya != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),Z) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Ya))),Ya) ) ) ).

% add_frac_num
tff(fact_2037_add__frac__num,axiom,
    ! [Ya: rat,Xc: rat,Z: rat] :
      ( ( Ya != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xc),Ya)),Z) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xc),aa_rat_rat(times_times_rat(Z),Ya))),Ya) ) ) ).

% add_frac_num
tff(fact_2038_add__frac__eq,axiom,
    ! [Ya: complex,Z: complex,Xc: complex,W: complex] :
      ( ( Ya != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xc),Ya)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xc),Z)),aa_complex_complex(times_times_complex(W),Ya))),aa_complex_complex(times_times_complex(Ya),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2039_add__frac__eq,axiom,
    ! [Ya: real,Z: real,Xc: real,W: real] :
      ( ( Ya != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2040_add__frac__eq,axiom,
    ! [Ya: rat,Z: rat,Xc: rat,W: rat] :
      ( ( Ya != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xc),Ya)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xc),Z)),aa_rat_rat(times_times_rat(W),Ya))),aa_rat_rat(times_times_rat(Ya),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2041_add__divide__eq__if__simps_I1_J,axiom,
    ! [A2: complex,B2: complex,Z: complex] :
      aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(divide1717551699836669952omplex(B2),Z)) = $ite(Z = zero_zero_complex,A2,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A2),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2042_add__divide__eq__if__simps_I1_J,axiom,
    ! [A2: real,B2: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),Z)) = $ite(Z = zero_zero_real,A2,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2043_add__divide__eq__if__simps_I1_J,axiom,
    ! [A2: rat,B2: rat,Z: rat] :
      aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(divide_divide_rat(B2),Z)) = $ite(Z = zero_zero_rat,A2,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2044_add__divide__eq__if__simps_I2_J,axiom,
    ! [A2: complex,Z: complex,B2: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A2),Z)),B2) = $ite(Z = zero_zero_complex,B2,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(times_times_complex(B2),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2045_add__divide__eq__if__simps_I2_J,axiom,
    ! [A2: real,Z: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),Z)),B2) = $ite(Z = zero_zero_real,B2,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2046_add__divide__eq__if__simps_I2_J,axiom,
    ! [A2: rat,Z: rat,B2: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(A2),Z)),B2) = $ite(Z = zero_zero_rat,B2,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(times_times_rat(B2),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2047_less__add__iff2,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),E)),D2)) ) ).

% less_add_iff2
tff(fact_2048_less__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(B2),A2)),E)),D2)) ) ).

% less_add_iff2
tff(fact_2049_less__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),E)),C2)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),E)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(B2),A2)),E)),D2)) ) ).

% less_add_iff2
tff(fact_2050_less__add__iff1,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),E)),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),E)),C2)),D2) ) ).

% less_add_iff1
tff(fact_2051_less__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(A2),E)),C2)),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(B2),E)),D2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(A2),B2)),E)),C2)),D2) ) ).

% less_add_iff1
tff(fact_2052_less__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),E)),C2)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),E)),D2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(minus_minus_int(A2),B2)),E)),C2)),D2) ) ).

% less_add_iff1
tff(fact_2053_divide__diff__eq__iff,axiom,
    ! [Z: complex,Xc: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xc),Z)),Ya) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Xc),aa_complex_complex(times_times_complex(Ya),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2054_divide__diff__eq__iff,axiom,
    ! [Z: real,Xc: real,Ya: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Z)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2055_divide__diff__eq__iff,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xc),Z)),Ya) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Xc),aa_rat_rat(times_times_rat(Ya),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_2056_diff__divide__eq__iff,axiom,
    ! [Z: complex,Xc: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(Xc),aa_complex_complex(divide1717551699836669952omplex(Ya),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xc),Z)),Ya)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2057_diff__divide__eq__iff,axiom,
    ! [Z: real,Xc: real,Ya: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Ya),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Z)),Ya)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2058_diff__divide__eq__iff,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(Xc),aa_rat_rat(divide_divide_rat(Ya),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xc),Z)),Ya)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_2059_diff__frac__eq,axiom,
    ! [Ya: complex,Z: complex,Xc: complex,W: complex] :
      ( ( Ya != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xc),Ya)),aa_complex_complex(divide1717551699836669952omplex(W),Z)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xc),Z)),aa_complex_complex(times_times_complex(W),Ya))),aa_complex_complex(times_times_complex(Ya),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2060_diff__frac__eq,axiom,
    ! [Ya: real,Z: real,Xc: real,W: real] :
      ( ( Ya != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2061_diff__frac__eq,axiom,
    ! [Ya: rat,Z: rat,Xc: rat,W: rat] :
      ( ( Ya != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xc),Ya)),aa_rat_rat(divide_divide_rat(W),Z)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xc),Z)),aa_rat_rat(times_times_rat(W),Ya))),aa_rat_rat(times_times_rat(Ya),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_2062_add__divide__eq__if__simps_I4_J,axiom,
    ! [A2: complex,B2: complex,Z: complex] :
      aa_complex_complex(minus_minus_complex(A2),aa_complex_complex(divide1717551699836669952omplex(B2),Z)) = $ite(Z = zero_zero_complex,A2,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A2),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_2063_add__divide__eq__if__simps_I4_J,axiom,
    ! [A2: real,B2: real,Z: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),Z)) = $ite(Z = zero_zero_real,A2,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_2064_add__divide__eq__if__simps_I4_J,axiom,
    ! [A2: rat,B2: rat,Z: rat] :
      aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(divide_divide_rat(B2),Z)) = $ite(Z = zero_zero_rat,A2,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(A2),Z)),B2)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_2065_td__gal__lt,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2)),B2) ) ) ).

% td_gal_lt
tff(fact_2066_divide__left__mono__neg,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2067_divide__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C2),A2)),aa_rat_rat(divide_divide_rat(C2),B2)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2068_mult__imp__le__div__pos,axiom,
    ! [Ya: real,Z: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Ya)),Xc)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2069_mult__imp__le__div__pos,axiom,
    ! [Ya: rat,Z: rat,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Ya)),Xc)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(divide_divide_rat(Xc),Ya)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2070_mult__imp__div__pos__le,axiom,
    ! [Ya: real,Xc: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Ya))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2071_mult__imp__div__pos__le,axiom,
    ! [Ya: rat,Xc: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_rat_rat(times_times_rat(Z),Ya))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xc),Ya)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2072_pos__le__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2) ) ) ).

% pos_le_divide_eq
tff(fact_2073_pos__le__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),B2) ) ) ).

% pos_le_divide_eq
tff(fact_2074_pos__divide__le__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ) ) ).

% pos_divide_le_eq
tff(fact_2075_pos__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C2)) ) ) ).

% pos_divide_le_eq
tff(fact_2076_neg__le__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ) ) ).

% neg_le_divide_eq
tff(fact_2077_neg__le__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C2)) ) ) ).

% neg_le_divide_eq
tff(fact_2078_neg__divide__le__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2) ) ) ).

% neg_divide_le_eq
tff(fact_2079_neg__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),B2) ) ) ).

% neg_divide_le_eq
tff(fact_2080_divide__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C2),B2)) ) ) ) ).

% divide_left_mono
tff(fact_2081_divide__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(times_times_rat(A2),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(C2),A2)),aa_rat_rat(divide_divide_rat(C2),B2)) ) ) ) ).

% divide_left_mono
tff(fact_2082_le__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)) ) ) ).

% le_divide_eq
tff(fact_2083_le__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(divide_divide_rat(B2),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_2084_divide__le__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),A2)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),B2),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)) ) ) ).

% divide_le_eq
tff(fact_2085_divide__le__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),C2)),A2)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)) ) ) ).

% divide_le_eq
tff(fact_2086_frac__le__eq,axiom,
    ! [Ya: real,Z: real,Xc: real,W: real] :
      ( ( Ya != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_2087_frac__le__eq,axiom,
    ! [Ya: rat,Z: rat,Xc: rat,W: rat] :
      ( ( Ya != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xc),Ya)),aa_rat_rat(divide_divide_rat(W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xc),Z)),aa_rat_rat(times_times_rat(W),Ya))),aa_rat_rat(times_times_rat(Ya),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_2088_frac__less__eq,axiom,
    ! [Ya: real,Z: real,Xc: real,W: real] :
      ( ( Ya != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Z))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_2089_frac__less__eq,axiom,
    ! [Ya: rat,Z: rat,Xc: rat,W: rat] :
      ( ( Ya != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xc),Ya)),aa_rat_rat(divide_divide_rat(W),Z))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xc),Z)),aa_rat_rat(times_times_rat(W),Ya))),aa_rat_rat(times_times_rat(Ya),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_2090_pos2,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% pos2
tff(fact_2091_td__gal,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)),A2)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2)) ) ) ).

% td_gal
tff(fact_2092_power__sub,axiom,
    ! [Nb: nat,M: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ) ) ) ).

% power_sub
tff(fact_2093_scaling__mono,axiom,
    ! [U: real,V: real,R3: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),V)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R3)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R3),S2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,V),U))),S2))),V) ) ) ) ).

% scaling_mono
tff(fact_2094_scaling__mono,axiom,
    ! [U: rat,V: rat,R3: rat,S2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),V)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),R3)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R3),S2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(U),aa_rat_rat(divide_divide_rat(aa_rat_rat(times_times_rat(R3),aa_rat_rat(minus_minus_rat(V),U))),S2))),V) ) ) ) ).

% scaling_mono
tff(fact_2095_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)) ) ) ).

% power_minus_is_div
tff(fact_2096_two__pow__div__gt__le,axiom,
    ! [V: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,V),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% two_pow_div_gt_le
tff(fact_2097_less__two__pow__divI,axiom,
    ! [Xc: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))) ) ) ).

% less_two_pow_divI
tff(fact_2098_less__two__pow__divD,axiom,
    ! [Xc: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ) ) ).

% less_two_pow_divD
tff(fact_2099_less__eq__option__Some,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( ord_le2843612097646854710et_nat(some_set_nat(Xc),some_set_nat(Ya))
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya) ) ).

% less_eq_option_Some
tff(fact_2100_less__eq__option__Some,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ord_le2406147912482264968on_rat(some_rat(Xc),some_rat(Ya))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya) ) ).

% less_eq_option_Some
tff(fact_2101_less__eq__option__Some,axiom,
    ! [Xc: num,Ya: num] :
      ( ord_le6622620407824499402on_num(some_num(Xc),some_num(Ya))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya) ) ).

% less_eq_option_Some
tff(fact_2102_less__eq__option__Some,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,some_nat(Xc)),some_nat(Ya))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya) ) ).

% less_eq_option_Some
tff(fact_2103_less__eq__option__Some,axiom,
    ! [Xc: int,Ya: int] :
      ( ord_le1736525451366464988on_int(some_int(Xc),some_int(Ya))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya) ) ).

% less_eq_option_Some
tff(fact_2104_nat__less__power__trans,axiom,
    ! [Nb: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_less_power_trans
tff(fact_2105_diff__add__zero,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) = zero_zero_nat ).

% diff_add_zero
tff(fact_2106_diff__gt__0__iff__gt,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ).

% diff_gt_0_iff_gt
tff(fact_2107_diff__gt__0__iff__gt,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ).

% diff_gt_0_iff_gt
tff(fact_2108_diff__gt__0__iff__gt,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(minus_minus_int(A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ).

% diff_gt_0_iff_gt
tff(fact_2109_diff__ge__0__iff__ge,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ).

% diff_ge_0_iff_ge
tff(fact_2110_diff__ge__0__iff__ge,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ).

% diff_ge_0_iff_ge
tff(fact_2111_diff__ge__0__iff__ge,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ).

% diff_ge_0_iff_ge
tff(fact_2112_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2113_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2114_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A2),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2115_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2116_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2117_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2118_less__add__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ).

% less_add_same_cancel2
tff(fact_2119_less__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(plus_plus_rat(B2),A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ).

% less_add_same_cancel2
tff(fact_2120_less__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ).

% less_add_same_cancel2
tff(fact_2121_less__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(plus_plus_int(B2),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ).

% less_add_same_cancel2
tff(fact_2122_less__add__same__cancel1,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ).

% less_add_same_cancel1
tff(fact_2123_less__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(plus_plus_rat(A2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ).

% less_add_same_cancel1
tff(fact_2124_less__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ).

% less_add_same_cancel1
tff(fact_2125_less__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(plus_plus_int(A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ).

% less_add_same_cancel1
tff(fact_2126_add__less__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_2127_add__less__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),B2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_2128_add__less__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_2129_add__less__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),B2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_2130_add__left__cancel,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_2131_add__left__cancel,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),B2) = aa_rat_rat(plus_plus_rat(A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_2132_add__left__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_2133_add__left__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( aa_int_int(plus_plus_int(A2),B2) = aa_int_int(plus_plus_int(A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add_left_cancel
tff(fact_2134_add__right__cancel,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_2135_add__right__cancel,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B2),A2) = aa_rat_rat(plus_plus_rat(C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_2136_add__right__cancel,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_2137_add__right__cancel,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( aa_int_int(plus_plus_int(B2),A2) = aa_int_int(plus_plus_int(C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add_right_cancel
tff(fact_2138_real__divide__square__eq,axiom,
    ! [R3: real,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),R3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),R3) ).

% real_divide_square_eq
tff(fact_2139_le__zero__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),zero_zero_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% le_zero_eq
tff(fact_2140_not__gr__zero,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% not_gr_zero
tff(fact_2141_add_Oright__neutral,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(A2),zero_zero_complex) = A2 ).

% add.right_neutral
tff(fact_2142_add_Oright__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% add.right_neutral
tff(fact_2143_add_Oright__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(A2),zero_zero_rat) = A2 ).

% add.right_neutral
tff(fact_2144_add_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% add.right_neutral
tff(fact_2145_add_Oright__neutral,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(A2),zero_zero_int) = A2 ).

% add.right_neutral
tff(fact_2146_double__zero__sym,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% double_zero_sym
tff(fact_2147_double__zero__sym,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = aa_rat_rat(plus_plus_rat(A2),A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_2148_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(plus_plus_int(A2),A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_2149_add__cancel__left__left,axiom,
    ! [B2: complex,A2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(B2),A2) = A2 )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_left
tff(fact_2150_add__cancel__left__left,axiom,
    ! [B2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) = A2 )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_2151_add__cancel__left__left,axiom,
    ! [B2: rat,A2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B2),A2) = A2 )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_2152_add__cancel__left__left,axiom,
    ! [B2: nat,A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) = A2 )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_2153_add__cancel__left__left,axiom,
    ! [B2: int,A2: int] :
      ( ( aa_int_int(plus_plus_int(B2),A2) = A2 )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_2154_add__cancel__left__right,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A2),B2) = A2 )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_right
tff(fact_2155_add__cancel__left__right,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = A2 )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_2156_add__cancel__left__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),B2) = A2 )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_2157_add__cancel__left__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = A2 )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_2158_add__cancel__left__right,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(plus_plus_int(A2),B2) = A2 )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_2159_add__cancel__right__left,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(B2),A2) )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_left
tff(fact_2160_add__cancel__right__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_2161_add__cancel__right__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(B2),A2) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_2162_add__cancel__right__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_2163_add__cancel__right__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(plus_plus_int(B2),A2) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_2164_add__cancel__right__right,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 = aa_complex_complex(plus_plus_complex(A2),B2) )
    <=> ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_right
tff(fact_2165_add__cancel__right__right,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_2166_add__cancel__right__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(plus_plus_rat(A2),B2) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_2167_add__cancel__right__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_2168_add__cancel__right__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(plus_plus_int(A2),B2) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_2169_add__eq__0__iff__both__eq__0,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Ya) = zero_zero_nat )
    <=> ( ( Xc = zero_zero_nat )
        & ( Ya = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_2170_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Ya) )
    <=> ( ( Xc = zero_zero_nat )
        & ( Ya = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_2171_add__0,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A2) = A2 ).

% add_0
tff(fact_2172_add__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% add_0
tff(fact_2173_add__0,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A2) = A2 ).

% add_0
tff(fact_2174_add__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A2) = A2 ).

% add_0
tff(fact_2175_add__0,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(zero_zero_int),A2) = A2 ).

% add_0
tff(fact_2176_add__le__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% add_le_cancel_left
tff(fact_2177_add__le__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% add_le_cancel_left
tff(fact_2178_add__le__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% add_le_cancel_left
tff(fact_2179_add__le__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C2),A2)),aa_int_int(plus_plus_int(C2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% add_le_cancel_left
tff(fact_2180_add__le__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% add_le_cancel_right
tff(fact_2181_add__le__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% add_le_cancel_right
tff(fact_2182_add__le__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% add_le_cancel_right
tff(fact_2183_add__le__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% add_le_cancel_right
tff(fact_2184_add__less__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% add_less_cancel_left
tff(fact_2185_add__less__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% add_less_cancel_left
tff(fact_2186_add__less__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% add_less_cancel_left
tff(fact_2187_add__less__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C2),A2)),aa_int_int(plus_plus_int(C2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% add_less_cancel_left
tff(fact_2188_add__less__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% add_less_cancel_right
tff(fact_2189_add__less__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% add_less_cancel_right
tff(fact_2190_add__less__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% add_less_cancel_right
tff(fact_2191_add__less__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),C2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% add_less_cancel_right
tff(fact_2192_diff__self,axiom,
    ! [A2: complex] : aa_complex_complex(minus_minus_complex(A2),A2) = zero_zero_complex ).

% diff_self
tff(fact_2193_diff__self,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),A2) = zero_zero_real ).

% diff_self
tff(fact_2194_diff__self,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),A2) = zero_zero_rat ).

% diff_self
tff(fact_2195_diff__self,axiom,
    ! [A2: int] : aa_int_int(minus_minus_int(A2),A2) = zero_zero_int ).

% diff_self
tff(fact_2196_diff__0__right,axiom,
    ! [A2: complex] : aa_complex_complex(minus_minus_complex(A2),zero_zero_complex) = A2 ).

% diff_0_right
tff(fact_2197_diff__0__right,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),zero_zero_real) = A2 ).

% diff_0_right
tff(fact_2198_diff__0__right,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),zero_zero_rat) = A2 ).

% diff_0_right
tff(fact_2199_diff__0__right,axiom,
    ! [A2: int] : aa_int_int(minus_minus_int(A2),zero_zero_int) = A2 ).

% diff_0_right
tff(fact_2200_zero__diff,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A2) = zero_zero_nat ).

% zero_diff
tff(fact_2201_diff__zero,axiom,
    ! [A2: complex] : aa_complex_complex(minus_minus_complex(A2),zero_zero_complex) = A2 ).

% diff_zero
tff(fact_2202_diff__zero,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),zero_zero_real) = A2 ).

% diff_zero
tff(fact_2203_diff__zero,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),zero_zero_rat) = A2 ).

% diff_zero
tff(fact_2204_diff__zero,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),zero_zero_nat) = A2 ).

% diff_zero
tff(fact_2205_diff__zero,axiom,
    ! [A2: int] : aa_int_int(minus_minus_int(A2),zero_zero_int) = A2 ).

% diff_zero
tff(fact_2206_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: complex] : aa_complex_complex(minus_minus_complex(A2),A2) = zero_zero_complex ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2207_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),A2) = zero_zero_real ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2208_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(A2),A2) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2209_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),A2) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2210_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: int] : aa_int_int(minus_minus_int(A2),A2) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_2211_add__diff__cancel,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),B2) = A2 ).

% add_diff_cancel
tff(fact_2212_add__diff__cancel,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),B2) = A2 ).

% add_diff_cancel
tff(fact_2213_add__diff__cancel,axiom,
    ! [A2: int,B2: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),B2)),B2) = A2 ).

% add_diff_cancel
tff(fact_2214_diff__add__cancel,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),B2) = A2 ).

% diff_add_cancel
tff(fact_2215_diff__add__cancel,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),B2) = A2 ).

% diff_add_cancel
tff(fact_2216_diff__add__cancel,axiom,
    ! [A2: int,B2: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A2),B2)),B2) = A2 ).

% diff_add_cancel
tff(fact_2217_add__diff__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) ).

% add_diff_cancel_left
tff(fact_2218_add__diff__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2)) = aa_rat_rat(minus_minus_rat(A2),B2) ).

% add_diff_cancel_left
tff(fact_2219_add__diff__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2) ).

% add_diff_cancel_left
tff(fact_2220_add__diff__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(C2),A2)),aa_int_int(plus_plus_int(C2),B2)) = aa_int_int(minus_minus_int(A2),B2) ).

% add_diff_cancel_left
tff(fact_2221_add__diff__cancel__left_H,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_2222_add__diff__cancel__left_H,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_2223_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_2224_add__diff__cancel__left_H,axiom,
    ! [A2: int,B2: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),B2)),A2) = B2 ).

% add_diff_cancel_left'
tff(fact_2225_add__diff__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) ).

% add_diff_cancel_right
tff(fact_2226_add__diff__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(A2),B2) ).

% add_diff_cancel_right
tff(fact_2227_add__diff__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2) ).

% add_diff_cancel_right
tff(fact_2228_add__diff__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),C2)) = aa_int_int(minus_minus_int(A2),B2) ).

% add_diff_cancel_right
tff(fact_2229_add__diff__cancel__right_H,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_2230_add__diff__cancel__right_H,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_2231_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_2232_add__diff__cancel__right_H,axiom,
    ! [A2: int,B2: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),B2)),B2) = A2 ).

% add_diff_cancel_right'
tff(fact_2233_less__option__Some,axiom,
    ! [Xc: real,Ya: real] :
      ( ord_less_option_real(some_real(Xc),some_real(Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ).

% less_option_Some
tff(fact_2234_less__option__Some,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ord_less_option_rat(some_rat(Xc),some_rat(Ya))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya) ) ).

% less_option_Some
tff(fact_2235_less__option__Some,axiom,
    ! [Xc: num,Ya: num] :
      ( ord_less_option_num(some_num(Xc),some_num(Ya))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya) ) ).

% less_option_Some
tff(fact_2236_less__option__Some,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,some_nat(Xc)),some_nat(Ya))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya) ) ).

% less_option_Some
tff(fact_2237_less__option__Some,axiom,
    ! [Xc: int,Ya: int] :
      ( ord_less_option_int(some_int(Xc),some_int(Ya))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya) ) ).

% less_option_Some
tff(fact_2238_add__le__same__cancel1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_2239_add__le__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(B2),A2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_2240_add__le__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_2241_add__le__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(B2),A2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_2242_add__le__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_2243_add__le__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),B2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_2244_add__le__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_2245_add__le__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),B2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_2246_le__add__same__cancel1,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ).

% le_add_same_cancel1
tff(fact_2247_le__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(plus_plus_rat(A2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ).

% le_add_same_cancel1
tff(fact_2248_le__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2) ) ).

% le_add_same_cancel1
tff(fact_2249_le__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(plus_plus_int(A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) ).

% le_add_same_cancel1
tff(fact_2250_le__add__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ).

% le_add_same_cancel2
tff(fact_2251_le__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(plus_plus_rat(B2),A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ).

% le_add_same_cancel2
tff(fact_2252_le__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2) ) ).

% le_add_same_cancel2
tff(fact_2253_le__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(plus_plus_int(B2),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) ).

% le_add_same_cancel2
tff(fact_2254_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2255_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2256_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2257_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2258_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2259_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(A2),A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2260_add__less__same__cancel1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_2261_add__less__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(B2),A2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_2262_add__less__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_2263_add__less__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(B2),A2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_2264_not__real__square__gt__zero,axiom,
    ! [Xc: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc))
    <=> ( Xc = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_2265_zero__reorient,axiom,
    ! [Xc: complex] :
      ( ( zero_zero_complex = Xc )
    <=> ( Xc = zero_zero_complex ) ) ).

% zero_reorient
tff(fact_2266_zero__reorient,axiom,
    ! [Xc: real] :
      ( ( zero_zero_real = Xc )
    <=> ( Xc = zero_zero_real ) ) ).

% zero_reorient
tff(fact_2267_zero__reorient,axiom,
    ! [Xc: rat] :
      ( ( zero_zero_rat = Xc )
    <=> ( Xc = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_2268_zero__reorient,axiom,
    ! [Xc: nat] :
      ( ( zero_zero_nat = Xc )
    <=> ( Xc = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_2269_zero__reorient,axiom,
    ! [Xc: int] :
      ( ( zero_zero_int = Xc )
    <=> ( Xc = zero_zero_int ) ) ).

% zero_reorient
tff(fact_2270_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2271_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( aa_rat_rat(plus_plus_rat(I),K) = aa_rat_rat(plus_plus_rat(J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2272_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2273_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( I = J2 )
        & ( K = L ) )
     => ( aa_int_int(plus_plus_int(I),K) = aa_int_int(plus_plus_int(J2),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2274_group__cancel_Oadd1,axiom,
    ! [A: real,K: real,A2: real,B2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% group_cancel.add1
tff(fact_2275_group__cancel_Oadd1,axiom,
    ! [A: rat,K: rat,A2: rat,B2: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(K),A2) )
     => ( aa_rat_rat(plus_plus_rat(A),B2) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A2),B2)) ) ) ).

% group_cancel.add1
tff(fact_2276_group__cancel_Oadd1,axiom,
    ! [A: nat,K: nat,A2: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% group_cancel.add1
tff(fact_2277_group__cancel_Oadd1,axiom,
    ! [A: int,K: int,A2: int,B2: int] :
      ( ( A = aa_int_int(plus_plus_int(K),A2) )
     => ( aa_int_int(plus_plus_int(A),B2) = aa_int_int(plus_plus_int(K),aa_int_int(plus_plus_int(A2),B2)) ) ) ).

% group_cancel.add1
tff(fact_2278_group__cancel_Oadd2,axiom,
    ! [B: real,K: real,B2: real,A2: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% group_cancel.add2
tff(fact_2279_group__cancel_Oadd2,axiom,
    ! [B: rat,K: rat,B2: rat,A2: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(K),B2) )
     => ( aa_rat_rat(plus_plus_rat(A2),B) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(plus_plus_rat(A2),B2)) ) ) ).

% group_cancel.add2
tff(fact_2280_group__cancel_Oadd2,axiom,
    ! [B: nat,K: nat,B2: nat,A2: nat] :
      ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% group_cancel.add2
tff(fact_2281_group__cancel_Oadd2,axiom,
    ! [B: int,K: int,B2: int,A2: int] :
      ( ( B = aa_int_int(plus_plus_int(K),B2) )
     => ( aa_int_int(plus_plus_int(A2),B) = aa_int_int(plus_plus_int(K),aa_int_int(plus_plus_int(A2),B2)) ) ) ).

% group_cancel.add2
tff(fact_2282_add_Oassoc,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% add.assoc
tff(fact_2283_add_Oassoc,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) = aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) ).

% add.assoc
tff(fact_2284_add_Oassoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ).

% add.assoc
tff(fact_2285_add_Oassoc,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A2),B2)),C2) = aa_int_int(plus_plus_int(A2),aa_int_int(plus_plus_int(B2),C2)) ).

% add.assoc
tff(fact_2286_add_Oleft__cancel,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add.left_cancel
tff(fact_2287_add_Oleft__cancel,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),B2) = aa_rat_rat(plus_plus_rat(A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add.left_cancel
tff(fact_2288_add_Oleft__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( aa_int_int(plus_plus_int(A2),B2) = aa_int_int(plus_plus_int(A2),C2) )
    <=> ( B2 = C2 ) ) ).

% add.left_cancel
tff(fact_2289_add_Oright__cancel,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add.right_cancel
tff(fact_2290_add_Oright__cancel,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B2),A2) = aa_rat_rat(plus_plus_rat(C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add.right_cancel
tff(fact_2291_add_Oright__cancel,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( aa_int_int(plus_plus_int(B2),A2) = aa_int_int(plus_plus_int(C2),A2) )
    <=> ( B2 = C2 ) ) ).

% add.right_cancel
tff(fact_2292_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_2293_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(plus_plus_rat(A2),B2) = aa_rat_rat(plus_plus_rat(B2),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_2294_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_2295_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A2: int,B2: int] : aa_int_int(plus_plus_int(A2),B2) = aa_int_int(plus_plus_int(B2),A2) ).

% ab_semigroup_add_class.add.commute
tff(fact_2296_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: real,A2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2297_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(B2),aa_rat_rat(plus_plus_rat(A2),C2)) = aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2298_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2299_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: int,A2: int,C2: int] : aa_int_int(plus_plus_int(B2),aa_int_int(plus_plus_int(A2),C2)) = aa_int_int(plus_plus_int(A2),aa_int_int(plus_plus_int(B2),C2)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2300_add__left__imp__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_2301_add__left__imp__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),B2) = aa_rat_rat(plus_plus_rat(A2),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_2302_add__left__imp__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_2303_add__left__imp__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( aa_int_int(plus_plus_int(A2),B2) = aa_int_int(plus_plus_int(A2),C2) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
tff(fact_2304_add__right__imp__eq,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_2305_add__right__imp__eq,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(B2),A2) = aa_rat_rat(plus_plus_rat(C2),A2) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_2306_add__right__imp__eq,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_2307_add__right__imp__eq,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( aa_int_int(plus_plus_int(B2),A2) = aa_int_int(plus_plus_int(C2),A2) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
tff(fact_2308_mult_Oassoc,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% mult.assoc
tff(fact_2309_mult_Oassoc,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A2),B2)),C2) = aa_rat_rat(times_times_rat(A2),aa_rat_rat(times_times_rat(B2),C2)) ).

% mult.assoc
tff(fact_2310_mult_Oassoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ).

% mult.assoc
tff(fact_2311_mult_Oassoc,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(times_times_int(aa_int_int(times_times_int(A2),B2)),C2) = aa_int_int(times_times_int(A2),aa_int_int(times_times_int(B2),C2)) ).

% mult.assoc
tff(fact_2312_mult_Oassoc,axiom,
    ! [A2: assn,B2: assn,C2: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),B2)),C2) = aa_assn_assn(times_times_assn(A2),aa_assn_assn(times_times_assn(B2),C2)) ).

% mult.assoc
tff(fact_2313_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_2314_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(A2),B2) = aa_rat_rat(times_times_rat(B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_2315_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_2316_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: int,B2: int] : aa_int_int(times_times_int(A2),B2) = aa_int_int(times_times_int(B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_2317_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(times_times_assn(A2),B2) = aa_assn_assn(times_times_assn(B2),A2) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_2318_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: real,A2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_2319_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C2: rat] : aa_rat_rat(times_times_rat(B2),aa_rat_rat(times_times_rat(A2),C2)) = aa_rat_rat(times_times_rat(A2),aa_rat_rat(times_times_rat(B2),C2)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_2320_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_2321_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: int,A2: int,C2: int] : aa_int_int(times_times_int(B2),aa_int_int(times_times_int(A2),C2)) = aa_int_int(times_times_int(A2),aa_int_int(times_times_int(B2),C2)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_2322_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C2: assn] : aa_assn_assn(times_times_assn(B2),aa_assn_assn(times_times_assn(A2),C2)) = aa_assn_assn(times_times_assn(A2),aa_assn_assn(times_times_assn(B2),C2)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_2323_diff__eq__diff__eq,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D2) )
     => ( ( A2 = B2 )
      <=> ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_2324_diff__eq__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B2) = aa_rat_rat(minus_minus_rat(C2),D2) )
     => ( ( A2 = B2 )
      <=> ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_2325_diff__eq__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( aa_int_int(minus_minus_int(A2),B2) = aa_int_int(minus_minus_int(C2),D2) )
     => ( ( A2 = B2 )
      <=> ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_2326_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: real,C2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2327_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: rat,C2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),C2)),B2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),C2) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2328_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),C2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),C2) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2329_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A2),C2)),B2) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A2),B2)),C2) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_2330_zero__le,axiom,
    ! [Xc: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xc) ).

% zero_le
tff(fact_2331_gr__zeroI,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% gr_zeroI
tff(fact_2332_not__less__zero,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less_zero
tff(fact_2333_gr__implies__not__zero,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_2334_zero__less__iff__neq__zero,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ( Nb != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
tff(fact_2335_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_2336_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_2337_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_2338_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_2339_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(zero_zero_int),A2) = A2 ).

% comm_monoid_add_class.add_0
tff(fact_2340_add_Ocomm__neutral,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(A2),zero_zero_complex) = A2 ).

% add.comm_neutral
tff(fact_2341_add_Ocomm__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% add.comm_neutral
tff(fact_2342_add_Ocomm__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(A2),zero_zero_rat) = A2 ).

% add.comm_neutral
tff(fact_2343_add_Ocomm__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% add.comm_neutral
tff(fact_2344_add_Ocomm__neutral,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(A2),zero_zero_int) = A2 ).

% add.comm_neutral
tff(fact_2345_add_Ogroup__left__neutral,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(zero_zero_complex),A2) = A2 ).

% add.group_left_neutral
tff(fact_2346_add_Ogroup__left__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A2) = A2 ).

% add.group_left_neutral
tff(fact_2347_add_Ogroup__left__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(zero_zero_rat),A2) = A2 ).

% add.group_left_neutral
tff(fact_2348_add_Ogroup__left__neutral,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(zero_zero_int),A2) = A2 ).

% add.group_left_neutral
tff(fact_2349_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J2)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2350_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J2)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J2),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2351_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2352_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J2)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2353_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( I = J2 )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2354_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( I = J2 )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J2),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2355_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I = J2 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2356_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( I = J2 )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2357_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J2)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2358_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J2),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2359_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2360_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2361_add__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2)) ) ) ).

% add_mono
tff(fact_2362_add__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),D2)) ) ) ).

% add_mono
tff(fact_2363_add__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2)) ) ) ).

% add_mono
tff(fact_2364_add__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),D2)) ) ) ).

% add_mono
tff(fact_2365_add__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)) ) ).

% add_left_mono
tff(fact_2366_add__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2)) ) ).

% add_left_mono
tff(fact_2367_add__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)) ) ).

% add_left_mono
tff(fact_2368_add__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C2),A2)),aa_int_int(plus_plus_int(C2),B2)) ) ).

% add_left_mono
tff(fact_2369_less__eqE,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ~ ! [C4: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C4) ) ).

% less_eqE
tff(fact_2370_add__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ) ).

% add_right_mono
tff(fact_2371_add__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2)) ) ).

% add_right_mono
tff(fact_2372_add__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ) ).

% add_right_mono
tff(fact_2373_add__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),C2)) ) ).

% add_right_mono
tff(fact_2374_le__iff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
    <=> ? [C5: nat] : B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C5) ) ).

% le_iff_add
tff(fact_2375_add__le__imp__le__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% add_le_imp_le_left
tff(fact_2376_add__le__imp__le__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% add_le_imp_le_left
tff(fact_2377_add__le__imp__le__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% add_le_imp_le_left
tff(fact_2378_add__le__imp__le__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(C2),A2)),aa_int_int(plus_plus_int(C2),B2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% add_le_imp_le_left
tff(fact_2379_add__le__imp__le__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% add_le_imp_le_right
tff(fact_2380_add__le__imp__le__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% add_le_imp_le_right
tff(fact_2381_add__le__imp__le__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% add_le_imp_le_right
tff(fact_2382_add__le__imp__le__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% add_le_imp_le_right
tff(fact_2383_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J2)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2384_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J2),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2385_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2386_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2387_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( ( I = J2 )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2388_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( ( I = J2 )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J2),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2389_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I = J2 )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2390_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( ( I = J2 )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2391_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J2)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2392_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J2)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J2),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2393_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2394_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J2)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2395_add__strict__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2)) ) ) ).

% add_strict_mono
tff(fact_2396_add__strict__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),D2)) ) ) ).

% add_strict_mono
tff(fact_2397_add__strict__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2)) ) ) ).

% add_strict_mono
tff(fact_2398_add__strict__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),D2)) ) ) ).

% add_strict_mono
tff(fact_2399_add__strict__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)) ) ).

% add_strict_left_mono
tff(fact_2400_add__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2)) ) ).

% add_strict_left_mono
tff(fact_2401_add__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)) ) ).

% add_strict_left_mono
tff(fact_2402_add__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C2),A2)),aa_int_int(plus_plus_int(C2),B2)) ) ).

% add_strict_left_mono
tff(fact_2403_add__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ) ).

% add_strict_right_mono
tff(fact_2404_add__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2)) ) ).

% add_strict_right_mono
tff(fact_2405_add__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ) ).

% add_strict_right_mono
tff(fact_2406_add__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),C2)) ) ).

% add_strict_right_mono
tff(fact_2407_add__less__imp__less__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% add_less_imp_less_left
tff(fact_2408_add__less__imp__less__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(C2),A2)),aa_rat_rat(plus_plus_rat(C2),B2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% add_less_imp_less_left
tff(fact_2409_add__less__imp__less__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% add_less_imp_less_left
tff(fact_2410_add__less__imp__less__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(C2),A2)),aa_int_int(plus_plus_int(C2),B2))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% add_less_imp_less_left
tff(fact_2411_add__less__imp__less__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% add_less_imp_less_right
tff(fact_2412_add__less__imp__less__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),C2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% add_less_imp_less_right
tff(fact_2413_add__less__imp__less__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% add_less_imp_less_right
tff(fact_2414_add__less__imp__less__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),C2))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% add_less_imp_less_right
tff(fact_2415_eq__iff__diff__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 = B2 )
    <=> ( aa_complex_complex(minus_minus_complex(A2),B2) = zero_zero_complex ) ) ).

% eq_iff_diff_eq_0
tff(fact_2416_eq__iff__diff__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = B2 )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_2417_eq__iff__diff__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
    <=> ( aa_rat_rat(minus_minus_rat(A2),B2) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_2418_eq__iff__diff__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( aa_int_int(minus_minus_int(A2),B2) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_2419_diff__mono,axiom,
    ! [A2: real,B2: real,D2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,D2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D2)) ) ) ).

% diff_mono
tff(fact_2420_diff__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),C2)),aa_rat_rat(minus_minus_rat(B2),D2)) ) ) ).

% diff_mono
tff(fact_2421_diff__mono,axiom,
    ! [A2: int,B2: int,D2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A2),C2)),aa_int_int(minus_minus_int(B2),D2)) ) ) ).

% diff_mono
tff(fact_2422_diff__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2)) ) ).

% diff_left_mono
tff(fact_2423_diff__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(C2),A2)),aa_rat_rat(minus_minus_rat(C2),B2)) ) ).

% diff_left_mono
tff(fact_2424_diff__left__mono,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(C2),A2)),aa_int_int(minus_minus_int(C2),B2)) ) ).

% diff_left_mono
tff(fact_2425_diff__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) ) ).

% diff_right_mono
tff(fact_2426_diff__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),C2)),aa_rat_rat(minus_minus_rat(B2),C2)) ) ).

% diff_right_mono
tff(fact_2427_diff__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A2),C2)),aa_int_int(minus_minus_int(B2),C2)) ) ).

% diff_right_mono
tff(fact_2428_diff__eq__diff__less__eq,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2429_diff__eq__diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B2) = aa_rat_rat(minus_minus_rat(C2),D2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2430_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( aa_int_int(minus_minus_int(A2),B2) = aa_int_int(minus_minus_int(C2),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_2431_diff__strict__mono,axiom,
    ! [A2: real,B2: real,D2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D2)) ) ) ).

% diff_strict_mono
tff(fact_2432_diff__strict__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,D2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),C2)),aa_rat_rat(minus_minus_rat(B2),D2)) ) ) ).

% diff_strict_mono
tff(fact_2433_diff__strict__mono,axiom,
    ! [A2: int,B2: int,D2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A2),C2)),aa_int_int(minus_minus_int(B2),D2)) ) ) ).

% diff_strict_mono
tff(fact_2434_diff__eq__diff__less,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D2) ) ) ).

% diff_eq_diff_less
tff(fact_2435_diff__eq__diff__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B2) = aa_rat_rat(minus_minus_rat(C2),D2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2) ) ) ).

% diff_eq_diff_less
tff(fact_2436_diff__eq__diff__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( aa_int_int(minus_minus_int(A2),B2) = aa_int_int(minus_minus_int(C2),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2) ) ) ).

% diff_eq_diff_less
tff(fact_2437_diff__strict__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2)) ) ).

% diff_strict_left_mono
tff(fact_2438_diff__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(C2),A2)),aa_rat_rat(minus_minus_rat(C2),B2)) ) ).

% diff_strict_left_mono
tff(fact_2439_diff__strict__left__mono,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(C2),A2)),aa_int_int(minus_minus_int(C2),B2)) ) ).

% diff_strict_left_mono
tff(fact_2440_diff__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) ) ).

% diff_strict_right_mono
tff(fact_2441_diff__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),C2)),aa_rat_rat(minus_minus_rat(B2),C2)) ) ).

% diff_strict_right_mono
tff(fact_2442_diff__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A2),C2)),aa_int_int(minus_minus_int(B2),C2)) ) ).

% diff_strict_right_mono
tff(fact_2443_group__cancel_Osub1,axiom,
    ! [A: real,K: real,A2: real,B2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) ) ) ).

% group_cancel.sub1
tff(fact_2444_group__cancel_Osub1,axiom,
    ! [A: rat,K: rat,A2: rat,B2: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(K),A2) )
     => ( aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(plus_plus_rat(K),aa_rat_rat(minus_minus_rat(A2),B2)) ) ) ).

% group_cancel.sub1
tff(fact_2445_group__cancel_Osub1,axiom,
    ! [A: int,K: int,A2: int,B2: int] :
      ( ( A = aa_int_int(plus_plus_int(K),A2) )
     => ( aa_int_int(minus_minus_int(A),B2) = aa_int_int(plus_plus_int(K),aa_int_int(minus_minus_int(A2),B2)) ) ) ).

% group_cancel.sub1
tff(fact_2446_diff__eq__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = C2 )
    <=> ( A2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2) ) ) ).

% diff_eq_eq
tff(fact_2447_diff__eq__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A2),B2) = C2 )
    <=> ( A2 = aa_rat_rat(plus_plus_rat(C2),B2) ) ) ).

% diff_eq_eq
tff(fact_2448_diff__eq__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( aa_int_int(minus_minus_int(A2),B2) = C2 )
    <=> ( A2 = aa_int_int(plus_plus_int(C2),B2) ) ) ).

% diff_eq_eq
tff(fact_2449_eq__diff__eq,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = C2 ) ) ).

% eq_diff_eq
tff(fact_2450_eq__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(minus_minus_rat(C2),B2) )
    <=> ( aa_rat_rat(plus_plus_rat(A2),B2) = C2 ) ) ).

% eq_diff_eq
tff(fact_2451_eq__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( A2 = aa_int_int(minus_minus_int(C2),B2) )
    <=> ( aa_int_int(plus_plus_int(A2),B2) = C2 ) ) ).

% eq_diff_eq
tff(fact_2452_add__diff__eq,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) ).

% add_diff_eq
tff(fact_2453_add__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(minus_minus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),C2) ).

% add_diff_eq
tff(fact_2454_add__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(plus_plus_int(A2),aa_int_int(minus_minus_int(B2),C2)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),B2)),C2) ).

% add_diff_eq
tff(fact_2455_diff__diff__eq2,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),B2) ).

% diff_diff_eq2
tff(fact_2456_diff__diff__eq2,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(minus_minus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),C2)),B2) ).

% diff_diff_eq2
tff(fact_2457_diff__diff__eq2,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(minus_minus_int(A2),aa_int_int(minus_minus_int(B2),C2)) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),C2)),B2) ).

% diff_diff_eq2
tff(fact_2458_diff__add__eq,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),B2) ).

% diff_add_eq
tff(fact_2459_diff__add__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),C2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),C2)),B2) ).

% diff_add_eq
tff(fact_2460_diff__add__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A2),B2)),C2) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),C2)),B2) ).

% diff_add_eq
tff(fact_2461_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_2462_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),C2)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_2463_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(minus_minus_int(A2),aa_int_int(plus_plus_int(B2),C2)) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A2),C2)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_2464_add__implies__diff,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2) = A2 )
     => ( C2 = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) ) ) ).

% add_implies_diff
tff(fact_2465_add__implies__diff,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(C2),B2) = A2 )
     => ( C2 = aa_rat_rat(minus_minus_rat(A2),B2) ) ) ).

% add_implies_diff
tff(fact_2466_add__implies__diff,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2) = A2 )
     => ( C2 = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2) ) ) ).

% add_implies_diff
tff(fact_2467_add__implies__diff,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( aa_int_int(plus_plus_int(C2),B2) = A2 )
     => ( C2 = aa_int_int(minus_minus_int(A2),B2) ) ) ).

% add_implies_diff
tff(fact_2468_diff__diff__eq,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ).

% diff_diff_eq
tff(fact_2469_diff__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),C2) = aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(plus_plus_rat(B2),C2)) ).

% diff_diff_eq
tff(fact_2470_diff__diff__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ).

% diff_diff_eq
tff(fact_2471_diff__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(A2),B2)),C2) = aa_int_int(minus_minus_int(A2),aa_int_int(plus_plus_int(B2),C2)) ).

% diff_diff_eq
tff(fact_2472_diff__diff__less,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),M)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),Nb) ) ) ).

% diff_diff_less
tff(fact_2473_add__decreasing,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),B2) ) ) ).

% add_decreasing
tff(fact_2474_add__decreasing,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),B2) ) ) ).

% add_decreasing
tff(fact_2475_add__decreasing,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),B2) ) ) ).

% add_decreasing
tff(fact_2476_add__decreasing,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C2)),B2) ) ) ).

% add_decreasing
tff(fact_2477_add__increasing,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)) ) ) ).

% add_increasing
tff(fact_2478_add__increasing,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(plus_plus_rat(A2),C2)) ) ) ).

% add_increasing
tff(fact_2479_add__increasing,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)) ) ) ).

% add_increasing
tff(fact_2480_add__increasing,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(plus_plus_int(A2),C2)) ) ) ).

% add_increasing
tff(fact_2481_add__decreasing2,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),B2) ) ) ).

% add_decreasing2
tff(fact_2482_add__decreasing2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),C2)),B2) ) ) ).

% add_decreasing2
tff(fact_2483_add__decreasing2,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),B2) ) ) ).

% add_decreasing2
tff(fact_2484_add__decreasing2,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),C2)),B2) ) ) ).

% add_decreasing2
tff(fact_2485_add__increasing2,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)) ) ) ).

% add_increasing2
tff(fact_2486_add__increasing2,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(plus_plus_rat(A2),C2)) ) ) ).

% add_increasing2
tff(fact_2487_add__increasing2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)) ) ) ).

% add_increasing2
tff(fact_2488_add__increasing2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(plus_plus_int(A2),C2)) ) ) ).

% add_increasing2
tff(fact_2489_add__nonneg__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_2490_add__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_2491_add__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_2492_add__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(A2),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_2493_add__nonpos__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_2494_add__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),B2)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_2495_add__nonpos__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_2496_add__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),B2)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_2497_add__nonneg__eq__0__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya) = zero_zero_real )
        <=> ( ( Xc = zero_zero_real )
            & ( Ya = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2498_add__nonneg__eq__0__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => ( ( aa_rat_rat(plus_plus_rat(Xc),Ya) = zero_zero_rat )
        <=> ( ( Xc = zero_zero_rat )
            & ( Ya = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2499_add__nonneg__eq__0__iff,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xc)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Ya)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Ya) = zero_zero_nat )
        <=> ( ( Xc = zero_zero_nat )
            & ( Ya = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2500_add__nonneg__eq__0__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => ( ( aa_int_int(plus_plus_int(Xc),Ya) = zero_zero_int )
        <=> ( ( Xc = zero_zero_int )
            & ( Ya = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2501_add__nonpos__eq__0__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya) = zero_zero_real )
        <=> ( ( Xc = zero_zero_real )
            & ( Ya = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2502_add__nonpos__eq__0__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),zero_zero_rat)
       => ( ( aa_rat_rat(plus_plus_rat(Xc),Ya) = zero_zero_rat )
        <=> ( ( Xc = zero_zero_rat )
            & ( Ya = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2503_add__nonpos__eq__0__iff,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Ya) = zero_zero_nat )
        <=> ( ( Xc = zero_zero_nat )
            & ( Ya = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2504_add__nonpos__eq__0__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),zero_zero_int)
       => ( ( aa_int_int(plus_plus_int(Xc),Ya) = zero_zero_int )
        <=> ( ( Xc = zero_zero_int )
            & ( Ya = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2505_add__neg__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_2506_add__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),B2)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_2507_add__neg__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_2508_add__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),B2)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_2509_add__pos__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% add_pos_pos
tff(fact_2510_add__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),B2)) ) ) ).

% add_pos_pos
tff(fact_2511_add__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% add_pos_pos
tff(fact_2512_add__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A2),B2)) ) ) ).

% add_pos_pos
tff(fact_2513_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ~ ! [C4: nat] :
            ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C4) )
           => ( C4 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_2514_pos__add__strict,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)) ) ) ).

% pos_add_strict
tff(fact_2515_pos__add__strict,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(plus_plus_rat(A2),C2)) ) ) ).

% pos_add_strict
tff(fact_2516_pos__add__strict,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)) ) ) ).

% pos_add_strict
tff(fact_2517_pos__add__strict,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(plus_plus_int(A2),C2)) ) ) ).

% pos_add_strict
tff(fact_2518_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J2)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2519_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J2),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2520_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2521_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2522_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J2: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J2)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J2),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2523_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J2: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(I),K)),aa_rat_rat(plus_plus_rat(J2),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2524_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J2: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J2),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2525_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J2: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(I),K)),aa_int_int(plus_plus_int(J2),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2526_add__le__less__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),D2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2)) ) ) ).

% add_le_less_mono
tff(fact_2527_add__le__less__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),D2)) ) ) ).

% add_le_less_mono
tff(fact_2528_add__le__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2)) ) ) ).

% add_le_less_mono
tff(fact_2529_add__le__less__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),D2)) ) ) ).

% add_le_less_mono
tff(fact_2530_add__less__le__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2)) ) ) ).

% add_less_le_mono
tff(fact_2531_add__less__le__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),D2)) ) ) ).

% add_less_le_mono
tff(fact_2532_add__less__le__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D2)) ) ) ).

% add_less_le_mono
tff(fact_2533_add__less__le__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),D2)) ) ) ).

% add_less_le_mono
tff(fact_2534_le__iff__diff__le__0,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_2535_le__iff__diff__le__0,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),B2)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_2536_le__iff__diff__le__0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A2),B2)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_2537_less__iff__diff__less__0,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_2538_less__iff__diff__less__0,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),B2)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_2539_less__iff__diff__less__0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A2),B2)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_2540_diff__le__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)) ) ).

% diff_le_eq
tff(fact_2541_diff__le__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),B2)),C2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(plus_plus_rat(C2),B2)) ) ).

% diff_le_eq
tff(fact_2542_diff__le__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A2),B2)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(plus_plus_int(C2),B2)) ) ).

% diff_le_eq
tff(fact_2543_le__diff__eq,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) ) ).

% le_diff_eq
tff(fact_2544_le__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(minus_minus_rat(C2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(A2),B2)),C2) ) ).

% le_diff_eq
tff(fact_2545_le__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(minus_minus_int(C2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),B2)),C2) ) ).

% le_diff_eq
tff(fact_2546_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),A2) = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_2547_le__add__diff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)),A2)) ) ).

% le_add_diff
tff(fact_2548_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),B2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_2549_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_2550_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),B2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_2551_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)),A2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_2552_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)),C2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_2553_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2)),B2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_2554_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2)) = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_2555_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2) = C2 )
        <=> ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C2),A2) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_2556_diff__less__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),C2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),B2)) ) ).

% diff_less_eq
tff(fact_2557_diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),B2)),C2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(plus_plus_rat(C2),B2)) ) ).

% diff_less_eq
tff(fact_2558_diff__less__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A2),B2)),C2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(plus_plus_int(C2),B2)) ) ).

% diff_less_eq
tff(fact_2559_less__diff__eq,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),C2) ) ).

% less_diff_eq
tff(fact_2560_less__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(minus_minus_rat(C2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),B2)),C2) ) ).

% less_diff_eq
tff(fact_2561_less__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(minus_minus_int(C2),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),B2)),C2) ) ).

% less_diff_eq
tff(fact_2562_add__neg__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_2563_add__neg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),B2)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_2564_add__neg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_2565_add__neg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),B2)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_2566_add__nonneg__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% add_nonneg_pos
tff(fact_2567_add__nonneg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),B2)) ) ) ).

% add_nonneg_pos
tff(fact_2568_add__nonneg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% add_nonneg_pos
tff(fact_2569_add__nonneg__pos,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A2),B2)) ) ) ).

% add_nonneg_pos
tff(fact_2570_add__nonpos__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_2571_add__nonpos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),B2)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_2572_add__nonpos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),zero_zero_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_2573_add__nonpos__neg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),B2)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_2574_add__pos__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ) ) ).

% add_pos_nonneg
tff(fact_2575_add__pos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(A2),B2)) ) ) ).

% add_pos_nonneg
tff(fact_2576_add__pos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% add_pos_nonneg
tff(fact_2577_add__pos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(A2),B2)) ) ) ).

% add_pos_nonneg
tff(fact_2578_add__strict__increasing,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)) ) ) ).

% add_strict_increasing
tff(fact_2579_add__strict__increasing,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(plus_plus_rat(A2),C2)) ) ) ).

% add_strict_increasing
tff(fact_2580_add__strict__increasing,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)) ) ) ).

% add_strict_increasing
tff(fact_2581_add__strict__increasing,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(plus_plus_int(A2),C2)) ) ) ).

% add_strict_increasing
tff(fact_2582_add__strict__increasing2,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2583_add__strict__increasing2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(plus_plus_rat(A2),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2584_add__strict__increasing2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2585_add__strict__increasing2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(plus_plus_int(A2),C2)) ) ) ).

% add_strict_increasing2
tff(fact_2586_n__less__equal__power__2,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% n_less_equal_power_2
tff(fact_2587_msrevs_I1_J,axiom,
    ! [Nb: nat,K: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),K) ) ) ).

% msrevs(1)
tff(fact_2588_nat__mult__power__less__eq,axiom,
    ! [B2: nat,A2: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))) ) ) ).

% nat_mult_power_less_eq
tff(fact_2589_nat__add__offset__less,axiom,
    ! [Ya: nat,Nb: nat,Xc: nat,M: nat,Sz: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
       => ( ( Sz = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Ya)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Sz)) ) ) ) ).

% nat_add_offset_less
tff(fact_2590_nat__power__less__diff,axiom,
    ! [Nb: nat,Q3: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))) ) ).

% nat_power_less_diff
tff(fact_2591_nat__le__power__trans,axiom,
    ! [Nb: nat,M: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% nat_le_power_trans
tff(fact_2592_real__average__minus__second,axiom,
    ! [B2: real,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_2593_real__average__minus__first,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_2594_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N: nat] :
            ( aa_nat_o(P,N)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)) ) )
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_2595_delt__out__of__range,axiom,
    ! [Xc: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xc) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya) ) ) ) ).

% delt_out_of_range
tff(fact_2596_div__exp__eq,axiom,
    ! [A2: code_integer,M: nat,Nb: nat] : divide6298287555418463151nteger(divide6298287555418463151nteger(A2,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = divide6298287555418463151nteger(A2,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ).

% div_exp_eq
tff(fact_2597_div__exp__eq,axiom,
    ! [A2: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ).

% div_exp_eq
tff(fact_2598_div__exp__eq,axiom,
    ! [A2: int,M: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ).

% div_exp_eq
tff(fact_2599_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,M: nat] :
      ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb) != zero_z3403309356797280102nteger )
     => ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2600_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2601_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2602_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Nb: nat] :
      ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_z3403309356797280102nteger )
     => ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2603_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2604_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2605_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Nb: nat] :
      ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_z3403309356797280102nteger )
     => ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb) != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2606_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2607_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2608_field__less__half__sum,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_2609_field__less__half__sum,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(Xc),Ya)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_2610_cnt__non__neg,axiom,
    ! [Ta: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Ta)) ).

% cnt_non_neg
tff(fact_2611_VEBT_Oinject_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12,X13a),X14a) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Y11,Y12,Y13),Y14) )
    <=> ( ( X11a = Y11 )
        & ( X12 = Y12 )
        & ( X13a = Y13 )
        & ( X14a = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_2612_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_2613_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% half_nonnegative_int_iff
tff(fact_2614_bits__div__by__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),zero_zero_nat) = zero_zero_nat ).

% bits_div_by_0
tff(fact_2615_bits__div__by__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_2616_bits__div__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_zero_nat),A2) = zero_zero_nat ).

% bits_div_0
tff(fact_2617_bits__div__0,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A2) = zero_zero_int ).

% bits_div_0
tff(fact_2618__C7_Oprems_C,axiom,
    aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary)),na) ).

% "7.prems"
tff(fact_2619_complete__real,axiom,
    ! [S: set_real] :
      ( ? [X4: real] : aa_set_real_o(member_real(X4),S)
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_set_real_o(member_real(X3),S)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z3) )
       => ? [Y: real] :
            ( ! [X4: real] :
                ( aa_set_real_o(member_real(X4),S)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y) )
            & ! [Z3: real] :
                ( ! [X3: real] :
                    ( aa_set_real_o(member_real(X3),S)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Z3) ) ) ) ) ).

% complete_real
tff(fact_2620_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D22)
       => ? [E2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2621_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D22)
       => ? [E2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2622_not__exp__less__eq__0__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),zero_zero_int) ).

% not_exp_less_eq_0_int
tff(fact_2623_add__diff__add,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),D2)) ).

% add_diff_add
tff(fact_2624_add__diff__add,axiom,
    ! [A2: rat,C2: rat,B2: rat,D2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),C2)),aa_rat_rat(plus_plus_rat(B2),D2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(A2),B2)),aa_rat_rat(minus_minus_rat(C2),D2)) ).

% add_diff_add
tff(fact_2625_add__diff__add,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] : aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),C2)),aa_int_int(plus_plus_int(B2),D2)) = aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(A2),B2)),aa_int_int(minus_minus_int(C2),D2)) ).

% add_diff_add
tff(fact_2626_less__eq__real__def,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% less_eq_real_def
tff(fact_2627_mult__diff__mult,axiom,
    ! [Xc: real,Ya: real,A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),B2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),A2)),B2)) ).

% mult_diff_mult
tff(fact_2628_mult__diff__mult,axiom,
    ! [Xc: rat,Ya: rat,A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xc),Ya)),aa_rat_rat(times_times_rat(A2),B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(Xc),aa_rat_rat(minus_minus_rat(Ya),B2))),aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(Xc),A2)),B2)) ).

% mult_diff_mult
tff(fact_2629_mult__diff__mult,axiom,
    ! [Xc: int,Ya: int,A2: int,B2: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(Xc),Ya)),aa_int_int(times_times_int(A2),B2)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Xc),aa_int_int(minus_minus_int(Ya),B2))),aa_int_int(times_times_int(aa_int_int(minus_minus_int(Xc),A2)),B2)) ).

% mult_diff_mult
tff(fact_2630_field__sum__of__halves,axiom,
    ! [Xc: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = Xc ).

% field_sum_of_halves
tff(fact_2631_field__sum__of__halves,axiom,
    ! [Xc: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(divide_divide_rat(Xc),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(Xc),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = Xc ).

% field_sum_of_halves
tff(fact_2632_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xc: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnodea: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xc = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnodea = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnodea) )
                     => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnodea)
                       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H)))),Maa))),Deg,Newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_2633_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
        & ( ( Xc = Mia )
          | ( Xc = Maa )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Maa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xc)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_2634_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mia: nat,Xc: nat,Maa: nat,Deg: nat,H: nat,L: nat,Newnodea: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xc)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnodea = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L) )
               => ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnodea)
                 => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnodea) )
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                     => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                  $ite(Xc = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H)))),Maa))),Deg,Newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_2635_nested__mint,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Vaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Nb)
     => ( ( Nb = aa_nat_nat(suc,aa_nat_nat(suc,Vaa)) )
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Mia)
         => ( ( Maa != Mia )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Vaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(vEBT_vebt_mint(Summarya))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Vaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(TreeLista)) ) ) ) ) ).

% nested_mint
tff(fact_2636_insert__simp__mima,axiom,
    ! [Xc: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xc = Mia )
        | ( Xc = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_2637_succ__min,axiom,
    ! [Deg: nat,Xc: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia)
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya),Xc) = some_nat(Mia) ) ) ) ).

% succ_min
tff(fact_2638_pred__max,axiom,
    ! [Deg: nat,Maa: nat,Xc: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xc)
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya),Xc) = some_nat(Maa) ) ) ) ).

% pred_max
tff(fact_2639_count__buildup,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb))) ).

% count_buildup
tff(fact_2640_mintlistlength,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Nb)
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Maa)
          & ? [M4: nat] :
              ( ( some_nat(M4) = vEBT_vebt_mint(Summarya) )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ) ) ).

% mintlistlength
tff(fact_2641_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_2642_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),zero_zero_nat) ).

% valid_tree_deg_neq_0
tff(fact_2643_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)),Nb)
     => ( Deg = Nb ) ) ).

% deg_deg_n
tff(fact_2644_delete__pres__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Ta),Xc)),Nb) ) ).

% delete_pres_valid
tff(fact_2645_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xc: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xc) ) ).

% min_Null_member
tff(fact_2646_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% deg_not_0
tff(fact_2647_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] : Tree = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList2),S3) ) ).

% deg_SUcn_Node
tff(fact_2648_dele__member__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Ta),Xc)),Ya)
      <=> ( ( Xc != Ya )
          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Ya) ) ) ) ).

% dele_member_cont_corr
tff(fact_2649_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb)),Nb) ) ).

% buildup_gives_valid
tff(fact_2650_mint__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Maxi) )
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Maxi) ) ) ).

% mint_member
tff(fact_2651_maxt__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Maxi) ) ) ).

% maxt_member
tff(fact_2652_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Mini) )
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xc)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),Xc) ) ) ) ).

% mint_corr_help
tff(fact_2653_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xc)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_2654_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xc: nat,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xc)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% member_bound
tff(fact_2655_misiz,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,M: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( some_nat(M) = vEBT_vebt_mint(Ta) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% misiz
tff(fact_2656_helpyd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xc) = some_nat(Ya) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% helpyd
tff(fact_2657_helpypredd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xc) = some_nat(Ya) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% helpypredd
tff(fact_2658_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Ta),Xc)),Ya)
           => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Ya)
              | ( Xc = Ya ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_2659_sumprop,axiom,
    aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,summary),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,na),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sumprop
tff(fact_2660_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)) ) ) ).

% mi_ma_2_deg
tff(fact_2661_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xc)
      <=> aa_set_nat_o(member_nat(Xc),vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_2662_summaxma,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Deg)
     => ( ( Mia != Maa )
       => ( the_nat(vEBT_vebt_maxt(Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_2663_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_2664_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_2665_zdiv__mult__self,axiom,
    ! [M: int,A2: int,Nb: int] :
      ( ( M != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A2),aa_int_int(times_times_int(M),Nb))),M) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),M)),Nb) ) ) ).

% zdiv_mult_self
tff(fact_2666_zdiv__le__dividend,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),A2) ) ) ).

% zdiv_le_dividend
tff(fact_2667_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(times_times_int(B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_2668_delete__correct_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( vEBT_VEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Ta),Xc)) = minus_minus_set_nat(vEBT_VEBT_set_vebt(Ta),insert_nat(Xc,bot_bot_set_nat)) ) ) ).

% delete_correct'
tff(fact_2669_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xc)
       => ( vEBT_vebt_maxt(Ta) = some_nat(Xc) ) ) ) ).

% maxt_sound
tff(fact_2670_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Xc) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xc) ) ) ).

% maxt_corr
tff(fact_2671_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Xc) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xc) ) ) ).

% mint_corr
tff(fact_2672_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xc)
       => ( vEBT_vebt_mint(Ta) = some_nat(Xc) ) ) ) ).

% mint_sound
tff(fact_2673_insert__simp__excp,axiom,
    ! [Mia: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Xc: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xc != Maa )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xc),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mia),Maa))),Deg,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summarya),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_2674_insert__simp__norm,axiom,
    ! [Xc: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xc)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( Xc != Maa )
           => ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),Maa))),Deg,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summarya),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_2675_pred__list__to__short,axiom,
    ! [Deg: nat,Xc: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Maa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya),Xc) = none_nat ) ) ) ) ).

% pred_list_to_short
tff(fact_2676_succ__list__to__short,axiom,
    ! [Deg: nat,Mia: nat,Xc: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xc)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya),Xc) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_2677_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta) ) ).

% minminNull
tff(fact_2678_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ( vEBT_vebt_mint(Ta) = none_nat ) ) ).

% minNullmin
tff(fact_2679_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_2680_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_2681_mint__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
tff(fact_2682_maxt__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = none_nat )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
tff(fact_2683_geqmaxNone,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Maa),Xc)
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya),Xc) = none_nat ) ) ) ).

% geqmaxNone
tff(fact_2684_delete__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( vEBT_VEBT_set_vebt(aa_nat_VEBT_VEBT(vEBT_vebt_delete(Ta),Xc)) = minus_minus_set_nat(vEBT_set_vebt(Ta),insert_nat(Xc,bot_bot_set_nat)) ) ) ).

% delete_correct
tff(fact_2685_max__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Nb)) ).

% max_Suc_Suc
tff(fact_2686_max__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),zero_zero_nat) = Nb ).

% max_0R
tff(fact_2687_max__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),Nb) = Nb ).

% max_0L
tff(fact_2688_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),zero_zero_nat) = A2 ).

% max_nat.right_neutral
tff(fact_2689_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) )
    <=> ( ( A2 = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_2690_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),A2) = A2 ).

% max_nat.left_neutral
tff(fact_2691_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = zero_zero_nat )
    <=> ( ( A2 = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_2692_less__eq__option__None__code,axiom,
    ! [Xc: option_nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,none_nat),Xc) ).

% less_eq_option_None_code
tff(fact_2693_less__option__None,axiom,
    ! [Xc: option_nat] : ~ aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,Xc),none_nat) ).

% less_option_None
tff(fact_2694_max__0__1_I3_J,axiom,
    ! [Xc: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),aa_num_real(numeral_numeral_real,Xc)) = aa_num_real(numeral_numeral_real,Xc) ).

% max_0_1(3)
tff(fact_2695_max__0__1_I3_J,axiom,
    ! [Xc: num] : aa_rat_rat(ord_max_rat(zero_zero_rat),aa_num_rat(numeral_numeral_rat,Xc)) = aa_num_rat(numeral_numeral_rat,Xc) ).

% max_0_1(3)
tff(fact_2696_max__0__1_I3_J,axiom,
    ! [Xc: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,Xc)) = aa_num_nat(numeral_numeral_nat,Xc) ).

% max_0_1(3)
tff(fact_2697_max__0__1_I3_J,axiom,
    ! [Xc: num] : aa_int_int(ord_max_int(zero_zero_int),aa_num_int(numeral_numeral_int,Xc)) = aa_num_int(numeral_numeral_int,Xc) ).

% max_0_1(3)
tff(fact_2698_max__0__1_I4_J,axiom,
    ! [Xc: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,Xc)),zero_zero_real) = aa_num_real(numeral_numeral_real,Xc) ).

% max_0_1(4)
tff(fact_2699_max__0__1_I4_J,axiom,
    ! [Xc: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xc)),zero_zero_rat) = aa_num_rat(numeral_numeral_rat,Xc) ).

% max_0_1(4)
tff(fact_2700_max__0__1_I4_J,axiom,
    ! [Xc: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xc)),zero_zero_nat) = aa_num_nat(numeral_numeral_nat,Xc) ).

% max_0_1(4)
tff(fact_2701_max__0__1_I4_J,axiom,
    ! [Xc: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,Xc)),zero_zero_int) = aa_num_int(numeral_numeral_int,Xc) ).

% max_0_1(4)
tff(fact_2702_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(1)
tff(fact_2703_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(1)
tff(fact_2704_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,U)),aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V),aa_num_nat(numeral_numeral_nat,U)) ).

% max_number_of(1)
tff(fact_2705_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(1)
tff(fact_2706_less__eq__option__Some__None,axiom,
    ! [Xc: nat] : ~ aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,some_nat(Xc)),none_nat) ).

% less_eq_option_Some_None
tff(fact_2707_less__option__None__Some__code,axiom,
    ! [Xc: nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),some_nat(Xc)) ).

% less_option_None_Some_code
tff(fact_2708_pred__member,axiom,
    ! [Ta: vEBT_VEBT,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xc),Ya)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Ya)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc)
        & ! [Z4: nat] :
            ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Z4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z4),Xc) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z4),Ya) ) ) ) ).

% pred_member
tff(fact_2709_succ__member,axiom,
    ! [Ta: vEBT_VEBT,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xc),Ya)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Ya)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
        & ! [Z4: nat] :
            ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Z4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Z4) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Z4) ) ) ) ).

% succ_member
tff(fact_2710_succ__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xc) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xc),Sx) ) ) ).

% succ_corr
tff(fact_2711_pred__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat,Px: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xc) = some_nat(Px) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta)),Xc),Px) ) ) ).

% pred_corr
tff(fact_2712_pred__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xc) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(vEBT_set_vebt(Ta)),Xc),Sx) ) ) ).

% pred_correct
tff(fact_2713_succ__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat,Sx: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xc) = some_nat(Sx) )
      <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_set_vebt(Ta)),Xc),Sx) ) ) ).

% succ_correct
tff(fact_2714_max__add__distrib__right,axiom,
    ! [Xc: real,Ya: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Ya),Z)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Z)) ).

% max_add_distrib_right
tff(fact_2715_max__add__distrib__right,axiom,
    ! [Xc: rat,Ya: rat,Z: rat] : aa_rat_rat(plus_plus_rat(Xc),aa_rat_rat(ord_max_rat(Ya),Z)) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xc),Ya)),aa_rat_rat(plus_plus_rat(Xc),Z)) ).

% max_add_distrib_right
tff(fact_2716_max__add__distrib__right,axiom,
    ! [Xc: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Ya),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Ya)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Z)) ).

% max_add_distrib_right
tff(fact_2717_max__add__distrib__right,axiom,
    ! [Xc: int,Ya: int,Z: int] : aa_int_int(plus_plus_int(Xc),aa_int_int(ord_max_int(Ya),Z)) = aa_int_int(ord_max_int(aa_int_int(plus_plus_int(Xc),Ya)),aa_int_int(plus_plus_int(Xc),Z)) ).

% max_add_distrib_right
tff(fact_2718_max__add__distrib__left,axiom,
    ! [Xc: real,Ya: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xc),Ya)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ya),Z)) ).

% max_add_distrib_left
tff(fact_2719_max__add__distrib__left,axiom,
    ! [Xc: rat,Ya: rat,Z: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(ord_max_rat(Xc),Ya)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(plus_plus_rat(Xc),Z)),aa_rat_rat(plus_plus_rat(Ya),Z)) ).

% max_add_distrib_left
tff(fact_2720_max__add__distrib__left,axiom,
    ! [Xc: nat,Ya: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),Ya)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ya),Z)) ).

% max_add_distrib_left
tff(fact_2721_max__add__distrib__left,axiom,
    ! [Xc: int,Ya: int,Z: int] : aa_int_int(plus_plus_int(aa_int_int(ord_max_int(Xc),Ya)),Z) = aa_int_int(ord_max_int(aa_int_int(plus_plus_int(Xc),Z)),aa_int_int(plus_plus_int(Ya),Z)) ).

% max_add_distrib_left
tff(fact_2722_max__diff__distrib__left,axiom,
    ! [Xc: real,Ya: real,Z: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xc),Ya)),Z) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),Z)) ).

% max_diff_distrib_left
tff(fact_2723_max__diff__distrib__left,axiom,
    ! [Xc: rat,Ya: rat,Z: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(ord_max_rat(Xc),Ya)),Z) = aa_rat_rat(ord_max_rat(aa_rat_rat(minus_minus_rat(Xc),Z)),aa_rat_rat(minus_minus_rat(Ya),Z)) ).

% max_diff_distrib_left
tff(fact_2724_max__diff__distrib__left,axiom,
    ! [Xc: int,Ya: int,Z: int] : aa_int_int(minus_minus_int(aa_int_int(ord_max_int(Xc),Ya)),Z) = aa_int_int(ord_max_int(aa_int_int(minus_minus_int(Xc),Z)),aa_int_int(minus_minus_int(Ya),Z)) ).

% max_diff_distrib_left
tff(fact_2725_nat__add__max__right,axiom,
    ! [M: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)) ).

% nat_add_max_right
tff(fact_2726_nat__add__max__left,axiom,
    ! [M: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q3)) ).

% nat_add_max_left
tff(fact_2727_nat__mult__max__right,axiom,
    ! [M: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_max_right
tff(fact_2728_nat__mult__max__left,axiom,
    ! [M: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,M),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ).

% nat_mult_max_left
tff(fact_2729_less__eq__option__None,axiom,
    ! [Xc: option_nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,none_nat),Xc) ).

% less_eq_option_None
tff(fact_2730_less__eq__option__None__is__None,axiom,
    ! [Xc: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_le5914376470875661696on_nat,Xc),none_nat)
     => ( Xc = none_nat ) ) ).

% less_eq_option_None_is_None
tff(fact_2731_nat__minus__add__max,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)),M) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),M) ).

% nat_minus_add_max
tff(fact_2732_less__option__None__Some,axiom,
    ! [Xc: nat] : aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),some_nat(Xc)) ).

% less_option_None_Some
tff(fact_2733_less__option__None__is__Some,axiom,
    ! [Xc: option_nat] :
      ( aa_option_nat_o(aa_opt4319133605541840243_nat_o(ord_less_option_nat,none_nat),Xc)
     => ? [Z2: nat] : Xc = some_nat(Z2) ) ).

% less_option_None_is_Some
tff(fact_2734_cnt__bound_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Ta)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)),one_one_real))) ) ).

% cnt_bound'
tff(fact_2735_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K),M)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_2736_cnt__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Ta)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one)))))))) ) ).

% cnt_bound
tff(fact_2737_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( some_nat(the_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_2738_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) = Option ) ) ).

% option.collapse
tff(fact_2739_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_2740_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),Xi)),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I3,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua))),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_2741_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),Xi)),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I3,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua))),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_2742_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),Xi)),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I3,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua))),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),C2,Q)
           => hoare_7629718768684598413on_nat(P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_2743_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua)),Xi)),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),vEBT_vebt_assn_raw,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I3,vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Uu),Uua))),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),C2,Q)
           => hoare_3067605981109127869le_nat(P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_2744_two__powr__height__bound__deg,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).

% two_powr_height_bound_deg
tff(fact_2745_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)),Xc) ) ) ) ).

% both_member_options_ding
tff(fact_2746_del__single__cont,axiom,
    ! [Xc: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xc = Mia )
        & ( Xc = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya) ) ) ) ).

% del_single_cont
tff(fact_2747_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),X_12) ) ).

% not_min_Null_member
tff(fact_2748_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xc: nat] :
      ( ( vEBT_vebt_maxt(Ta) = some_nat(Xc) )
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xc) ) ).

% maxbmo
tff(fact_2749_dele__bmo__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Ta),Xc)),Ya)
      <=> ( ( Xc != Ya )
          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Ya) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_2750_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xc)
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xc) ) ) ).

% valid_member_both_member_options
tff(fact_2751_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xc)
      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta),Xc) ) ) ).

% both_member_options_equiv_member
tff(fact_2752_semiring__norm_I90_J,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_num_num(bit1,M) = aa_num_num(bit1,Nb) )
    <=> ( M = Nb ) ) ).

% semiring_norm(90)
tff(fact_2753_option_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( some_nat(X22) = some_nat(Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_2754_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( some_P7363390416028606310at_nat(X22) = some_P7363390416028606310at_nat(Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_2755_mult_Oright__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),one_one_real) = A2 ).

% mult.right_neutral
tff(fact_2756_mult_Oright__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),one_one_rat) = A2 ).

% mult.right_neutral
tff(fact_2757_mult_Oright__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),one_one_nat) = A2 ).

% mult.right_neutral
tff(fact_2758_mult_Oright__neutral,axiom,
    ! [A2: int] : aa_int_int(times_times_int(A2),one_one_int) = A2 ).

% mult.right_neutral
tff(fact_2759_mult_Oright__neutral,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(A2),one_one_assn) = A2 ).

% mult.right_neutral
tff(fact_2760_mult__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A2) = A2 ).

% mult_1
tff(fact_2761_mult__1,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(one_one_rat),A2) = A2 ).

% mult_1
tff(fact_2762_mult__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A2) = A2 ).

% mult_1
tff(fact_2763_mult__1,axiom,
    ! [A2: int] : aa_int_int(times_times_int(one_one_int),A2) = A2 ).

% mult_1
tff(fact_2764_mult__1,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(one_one_assn),A2) = A2 ).

% mult_1
tff(fact_2765_div__by__1,axiom,
    ! [A2: complex] : aa_complex_complex(divide1717551699836669952omplex(A2),one_one_complex) = A2 ).

% div_by_1
tff(fact_2766_div__by__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),one_one_real) = A2 ).

% div_by_1
tff(fact_2767_div__by__1,axiom,
    ! [A2: rat] : aa_rat_rat(divide_divide_rat(A2),one_one_rat) = A2 ).

% div_by_1
tff(fact_2768_div__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),one_one_nat) = A2 ).

% div_by_1
tff(fact_2769_div__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),one_one_int) = A2 ).

% div_by_1
tff(fact_2770_bits__div__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),one_one_nat) = A2 ).

% bits_div_by_1
tff(fact_2771_bits__div__by__1,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),one_one_int) = A2 ).

% bits_div_by_1
tff(fact_2772_power__one,axiom,
    ! [Nb: nat] : aa_nat_assn(power_power_assn(one_one_assn),Nb) = one_one_assn ).

% power_one
tff(fact_2773_power__one,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).

% power_one
tff(fact_2774_power__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),Nb) = one_one_nat ).

% power_one
tff(fact_2775_power__one,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),Nb) = one_one_real ).

% power_one
tff(fact_2776_power__one,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(one_one_int),Nb) = one_one_int ).

% power_one
tff(fact_2777_power__one,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(one_one_complex),Nb) = one_one_complex ).

% power_one
tff(fact_2778_power__one,axiom,
    ! [Nb: nat] : power_8256067586552552935nteger(one_one_Code_integer,Nb) = one_one_Code_integer ).

% power_one
tff(fact_2779_semiring__norm_I89_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(bit1,M) != aa_num_num(bit0,Nb) ).

% semiring_norm(89)
tff(fact_2780_semiring__norm_I88_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(bit0,M) != aa_num_num(bit1,Nb) ).

% semiring_norm(88)
tff(fact_2781_semiring__norm_I86_J,axiom,
    ! [M: num] : aa_num_num(bit1,M) != one ).

% semiring_norm(86)
tff(fact_2782_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit1,Nb) ).

% semiring_norm(84)
tff(fact_2783_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Ta),Xc)),Xc) ) ) ).

% valid_insert_both_member_options_add
tff(fact_2784_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xc)
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Ta),Ya)),Xc) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_2785_not__None__eq,axiom,
    ! [Xc: option_nat] :
      ( ( Xc != none_nat )
    <=> ? [Y4: nat] : Xc = some_nat(Y4) ) ).

% not_None_eq
tff(fact_2786_not__None__eq,axiom,
    ! [Xc: option4927543243414619207at_nat] :
      ( ( Xc != none_P5556105721700978146at_nat )
    <=> ? [Y4: product_prod_nat_nat] : Xc = some_P7363390416028606310at_nat(Y4) ) ).

% not_None_eq
tff(fact_2787_not__Some__eq,axiom,
    ! [Xc: option_nat] :
      ( ! [Y4: nat] : Xc != some_nat(Y4)
    <=> ( Xc = none_nat ) ) ).

% not_Some_eq
tff(fact_2788_not__Some__eq,axiom,
    ! [Xc: option4927543243414619207at_nat] :
      ( ! [Y4: product_prod_nat_nat] : Xc != some_P7363390416028606310at_nat(Y4)
    <=> ( Xc = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
tff(fact_2789_semiring__norm_I80_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb) ) ).

% semiring_norm(80)
tff(fact_2790_semiring__norm_I73_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Nb) ) ).

% semiring_norm(73)
tff(fact_2791_mult__cancel__left1,axiom,
    ! [C2: complex,B2: complex] :
      ( ( C2 = aa_complex_complex(times_times_complex(C2),B2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_2792_mult__cancel__left1,axiom,
    ! [C2: real,B2: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) )
    <=> ( ( C2 = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_2793_mult__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( C2 = aa_rat_rat(times_times_rat(C2),B2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_2794_mult__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( ( C2 = aa_int_int(times_times_int(C2),B2) )
    <=> ( ( C2 = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_2795_mult__cancel__left2,axiom,
    ! [C2: complex,A2: complex] :
      ( ( aa_complex_complex(times_times_complex(C2),A2) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = one_one_complex ) ) ) ).

% mult_cancel_left2
tff(fact_2796_mult__cancel__left2,axiom,
    ! [C2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_left2
tff(fact_2797_mult__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( ( aa_rat_rat(times_times_rat(C2),A2) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_2798_mult__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( ( aa_int_int(times_times_int(C2),A2) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_2799_mult__cancel__right1,axiom,
    ! [C2: complex,B2: complex] :
      ( ( C2 = aa_complex_complex(times_times_complex(B2),C2) )
    <=> ( ( C2 = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_2800_mult__cancel__right1,axiom,
    ! [C2: real,B2: real] :
      ( ( C2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2) )
    <=> ( ( C2 = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_2801_mult__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( C2 = aa_rat_rat(times_times_rat(B2),C2) )
    <=> ( ( C2 = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_2802_mult__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( ( C2 = aa_int_int(times_times_int(B2),C2) )
    <=> ( ( C2 = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_2803_mult__cancel__right2,axiom,
    ! [A2: complex,C2: complex] :
      ( ( aa_complex_complex(times_times_complex(A2),C2) = C2 )
    <=> ( ( C2 = zero_zero_complex )
        | ( A2 = one_one_complex ) ) ) ).

% mult_cancel_right2
tff(fact_2804_mult__cancel__right2,axiom,
    ! [A2: real,C2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = C2 )
    <=> ( ( C2 = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_right2
tff(fact_2805_mult__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),C2) = C2 )
    <=> ( ( C2 = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_2806_mult__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( ( aa_int_int(times_times_int(A2),C2) = C2 )
    <=> ( ( C2 = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_2807_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numera6690914467698888265omplex(Nb) = one_one_complex )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_2808_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Nb) = one_one_real )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_2809_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_rat(numeral_numeral_rat,Nb) = one_one_rat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_2810_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Nb) = one_one_nat )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_2811_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Nb) = one_one_int )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_2812_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_2813_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_real = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_2814_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_rat = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_2815_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_nat = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_2816_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_int = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_2817_diff__numeral__special_I9_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),one_one_complex) = zero_zero_complex ).

% diff_numeral_special(9)
tff(fact_2818_diff__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),one_one_real) = zero_zero_real ).

% diff_numeral_special(9)
tff(fact_2819_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_2820_diff__numeral__special_I9_J,axiom,
    aa_int_int(minus_minus_int(one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_2821_divide__eq__1__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A2),B2) = one_one_complex )
    <=> ( ( B2 != zero_zero_complex )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_2822_divide__eq__1__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) = one_one_real )
    <=> ( ( B2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_2823_divide__eq__1__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A2),B2) = one_one_rat )
    <=> ( ( B2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_2824_div__self,axiom,
    ! [A2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A2),A2) = one_one_complex ) ) ).

% div_self
tff(fact_2825_div__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),A2) = one_one_real ) ) ).

% div_self
tff(fact_2826_div__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A2),A2) = one_one_rat ) ) ).

% div_self
tff(fact_2827_div__self,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),A2) = one_one_nat ) ) ).

% div_self
tff(fact_2828_div__self,axiom,
    ! [A2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),A2) = one_one_int ) ) ).

% div_self
tff(fact_2829_one__eq__divide__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A2),B2) )
    <=> ( ( B2 != zero_zero_complex )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_2830_one__eq__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) )
    <=> ( ( B2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_2831_one__eq__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A2),B2) )
    <=> ( ( B2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_2832_divide__self,axiom,
    ! [A2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A2),A2) = one_one_complex ) ) ).

% divide_self
tff(fact_2833_divide__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),A2) = one_one_real ) ) ).

% divide_self
tff(fact_2834_divide__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A2),A2) = one_one_rat ) ) ).

% divide_self
tff(fact_2835_divide__self__if,axiom,
    ! [A2: complex] :
      aa_complex_complex(divide1717551699836669952omplex(A2),A2) = $ite(A2 = zero_zero_complex,zero_zero_complex,one_one_complex) ).

% divide_self_if
tff(fact_2836_divide__self__if,axiom,
    ! [A2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),A2) = $ite(A2 = zero_zero_real,zero_zero_real,one_one_real) ).

% divide_self_if
tff(fact_2837_divide__self__if,axiom,
    ! [A2: rat] :
      aa_rat_rat(divide_divide_rat(A2),A2) = $ite(A2 = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_2838_divide__eq__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2) = one_one_real )
    <=> ( ( A2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% divide_eq_eq_1
tff(fact_2839_divide__eq__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B2),A2) = one_one_rat )
    <=> ( ( A2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% divide_eq_eq_1
tff(fact_2840_eq__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2) )
    <=> ( ( A2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% eq_divide_eq_1
tff(fact_2841_eq__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B2),A2) )
    <=> ( ( A2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% eq_divide_eq_1
tff(fact_2842_one__divide__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% one_divide_eq_0_iff
tff(fact_2843_one__divide__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(one_one_rat),A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2844_zero__eq__1__divide__iff,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
tff(fact_2845_zero__eq__1__divide__iff,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = aa_rat_rat(divide_divide_rat(one_one_rat),A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2846_power__inject__exp,axiom,
    ! [A2: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( ( power_8256067586552552935nteger(A2,M) = power_8256067586552552935nteger(A2,Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_2847_power__inject__exp,axiom,
    ! [A2: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_2848_power__inject__exp,axiom,
    ! [A2: rat,M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( ( aa_nat_rat(power_power_rat(A2),M) = aa_nat_rat(power_power_rat(A2),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_2849_power__inject__exp,axiom,
    ! [A2: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_2850_power__inject__exp,axiom,
    ! [A2: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( ( aa_nat_int(power_power_int(A2),M) = aa_nat_int(power_power_int(A2),Nb) )
      <=> ( M = Nb ) ) ) ).

% power_inject_exp
tff(fact_2851_max__0__1_I1_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,zero_zero_real),one_one_real) = one_one_real ).

% max_0_1(1)
tff(fact_2852_max__0__1_I1_J,axiom,
    aa_rat_rat(ord_max_rat(zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_2853_max__0__1_I1_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),one_one_nat) = one_one_nat ).

% max_0_1(1)
tff(fact_2854_max__0__1_I1_J,axiom,
    aa_int_int(ord_max_int(zero_zero_int),one_one_int) = one_one_int ).

% max_0_1(1)
tff(fact_2855_max__0__1_I2_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),zero_zero_real) = one_one_real ).

% max_0_1(2)
tff(fact_2856_max__0__1_I2_J,axiom,
    aa_rat_rat(ord_max_rat(one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_2857_max__0__1_I2_J,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),zero_zero_nat) = one_one_nat ).

% max_0_1(2)
tff(fact_2858_max__0__1_I2_J,axiom,
    aa_int_int(ord_max_int(one_one_int),zero_zero_int) = one_one_int ).

% max_0_1(2)
tff(fact_2859_max__0__1_I5_J,axiom,
    ! [Xc: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,one_one_real),aa_num_real(numeral_numeral_real,Xc)) = aa_num_real(numeral_numeral_real,Xc) ).

% max_0_1(5)
tff(fact_2860_max__0__1_I5_J,axiom,
    ! [Xc: num] : aa_rat_rat(ord_max_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xc)) = aa_num_rat(numeral_numeral_rat,Xc) ).

% max_0_1(5)
tff(fact_2861_max__0__1_I5_J,axiom,
    ! [Xc: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xc)) = aa_num_nat(numeral_numeral_nat,Xc) ).

% max_0_1(5)
tff(fact_2862_max__0__1_I5_J,axiom,
    ! [Xc: num] : aa_int_int(ord_max_int(one_one_int),aa_num_int(numeral_numeral_int,Xc)) = aa_num_int(numeral_numeral_int,Xc) ).

% max_0_1(5)
tff(fact_2863_max__0__1_I6_J,axiom,
    ! [Xc: num] : aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,Xc)),one_one_real) = aa_num_real(numeral_numeral_real,Xc) ).

% max_0_1(6)
tff(fact_2864_max__0__1_I6_J,axiom,
    ! [Xc: num] : aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,Xc)),one_one_rat) = aa_num_rat(numeral_numeral_rat,Xc) ).

% max_0_1(6)
tff(fact_2865_max__0__1_I6_J,axiom,
    ! [Xc: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,Xc)),one_one_nat) = aa_num_nat(numeral_numeral_nat,Xc) ).

% max_0_1(6)
tff(fact_2866_max__0__1_I6_J,axiom,
    ! [Xc: num] : aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,Xc)),one_one_int) = aa_num_int(numeral_numeral_int,Xc) ).

% max_0_1(6)
tff(fact_2867_not__Some__eq2,axiom,
    ! [V: option2651255830984564193nteger] :
      ( ! [X2: code_integer,Y4: code_integer] : V != some_P6772290148444788224nteger(produc1086072967326762835nteger(X2,Y4))
    <=> ( V = none_P4506660739021792380nteger ) ) ).

% not_Some_eq2
tff(fact_2868_not__Some__eq2,axiom,
    ! [V: option936205604648967762et_nat] :
      ( ! [X2: heap_e7401611519738050253t_unit,Y4: set_nat] : V != some_P624177172695371229et_nat(produc7507926704131184380et_nat(X2,Y4))
    <=> ( V = none_P533106815845188193et_nat ) ) ).

% not_Some_eq2
tff(fact_2869_not__Some__eq2,axiom,
    ! [V: option2661157926820139483um_num] :
      ( ! [X2: num,Y4: num] : V != some_P6201964756284913402um_num(product_Pair_num_num(X2,Y4))
    <=> ( V = none_P4394680061957285238um_num ) ) ).

% not_Some_eq2
tff(fact_2870_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ! [X2: int,Y4: int] : V != some_P4184893108420464158nt_int(product_Pair_int_int(X2,Y4))
    <=> ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
tff(fact_2871_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ! [X2: nat,Y4: nat] : V != some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X2),Y4))
    <=> ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
tff(fact_2872_semiring__norm_I7_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(M),Nb)) ).

% semiring_norm(7)
tff(fact_2873_semiring__norm_I9_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(M),Nb)) ).

% semiring_norm(9)
tff(fact_2874_semiring__norm_I14_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,M)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(times_times_num(M),aa_num_num(bit1,Nb))) ).

% semiring_norm(14)
tff(fact_2875_semiring__norm_I15_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit1,M)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(times_times_num(aa_num_num(bit1,M)),Nb)) ).

% semiring_norm(15)
tff(fact_2876_semiring__norm_I72_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Nb) ) ).

% semiring_norm(72)
tff(fact_2877_semiring__norm_I81_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,M)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb) ) ).

% semiring_norm(81)
tff(fact_2878_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),one) ).

% semiring_norm(70)
tff(fact_2879_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa_num_o(aa_num_fun_num_o(ord_less_num,one),aa_num_num(bit1,Nb)) ).

% semiring_norm(77)
tff(fact_2880_divide__le__0__1__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% divide_le_0_1_iff
tff(fact_2881_divide__le__0__1__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_2882_zero__le__divide__1__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% zero_le_divide_1_iff
tff(fact_2883_zero__le__divide__1__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% zero_le_divide_1_iff
tff(fact_2884_divide__less__0__1__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% divide_less_0_1_iff
tff(fact_2885_divide__less__0__1__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_2886_divide__less__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% divide_less_eq_1_neg
tff(fact_2887_divide__less__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% divide_less_eq_1_neg
tff(fact_2888_divide__less__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ).

% divide_less_eq_1_pos
tff(fact_2889_divide__less__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ).

% divide_less_eq_1_pos
tff(fact_2890_less__divide__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ).

% less_divide_eq_1_neg
tff(fact_2891_less__divide__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ).

% less_divide_eq_1_neg
tff(fact_2892_less__divide__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% less_divide_eq_1_pos
tff(fact_2893_less__divide__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% less_divide_eq_1_pos
tff(fact_2894_zero__less__divide__1__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ).

% zero_less_divide_1_iff
tff(fact_2895_zero__less__divide__1__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(one_one_rat),A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% zero_less_divide_1_iff
tff(fact_2896_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A2),aa_complex_complex(times_times_complex(A2),B2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2897_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2898_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(times_times_rat(A2),B2)) = aa_rat_rat(divide_divide_rat(one_one_rat),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2899_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B2),aa_complex_complex(times_times_complex(A2),B2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A2) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2900_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2901_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B2),aa_rat_rat(times_times_rat(A2),B2)) = aa_rat_rat(divide_divide_rat(one_one_rat),A2) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2902_power__strict__increasing__iff,axiom,
    ! [B2: code_integer,Xc: nat,Ya: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(B2,Xc)),power_8256067586552552935nteger(B2,Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya) ) ) ).

% power_strict_increasing_iff
tff(fact_2903_power__strict__increasing__iff,axiom,
    ! [B2: real,Xc: nat,Ya: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Xc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya) ) ) ).

% power_strict_increasing_iff
tff(fact_2904_power__strict__increasing__iff,axiom,
    ! [B2: rat,Xc: nat,Ya: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B2),Xc)),aa_nat_rat(power_power_rat(B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya) ) ) ).

% power_strict_increasing_iff
tff(fact_2905_power__strict__increasing__iff,axiom,
    ! [B2: nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya) ) ) ).

% power_strict_increasing_iff
tff(fact_2906_power__strict__increasing__iff,axiom,
    ! [B2: int,Xc: nat,Ya: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),Xc)),aa_nat_int(power_power_int(B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya) ) ) ).

% power_strict_increasing_iff
tff(fact_2907_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_2908_semiring__norm_I10_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(M),Nb)),one)) ).

% semiring_norm(10)
tff(fact_2909_semiring__norm_I8_J,axiom,
    ! [M: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,M)),one) = aa_num_num(bit0,aa_num_num(plus_plus_num(M),one)) ).

% semiring_norm(8)
tff(fact_2910_semiring__norm_I5_J,axiom,
    ! [M: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,M)),one) = aa_num_num(bit1,M) ).

% semiring_norm(5)
tff(fact_2911_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(Nb),one)) ).

% semiring_norm(4)
tff(fact_2912_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit0,Nb)) = aa_num_num(bit1,Nb) ).

% semiring_norm(3)
tff(fact_2913_semiring__norm_I16_J,axiom,
    ! [M: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit1,M)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(M),Nb)),aa_num_num(bit0,aa_num_num(times_times_num(M),Nb)))) ).

% semiring_norm(16)
tff(fact_2914_semiring__norm_I79_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,M)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Nb) ) ).

% semiring_norm(79)
tff(fact_2915_semiring__norm_I74_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,M)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb) ) ).

% semiring_norm(74)
tff(fact_2916_divide__le__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% divide_le_eq_1_neg
tff(fact_2917_divide__le__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% divide_le_eq_1_neg
tff(fact_2918_divide__le__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ).

% divide_le_eq_1_pos
tff(fact_2919_divide__le__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ).

% divide_le_eq_1_pos
tff(fact_2920_le__divide__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ).

% le_divide_eq_1_neg
tff(fact_2921_le__divide__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ).

% le_divide_eq_1_neg
tff(fact_2922_le__divide__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% le_divide_eq_1_pos
tff(fact_2923_le__divide__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% le_divide_eq_1_pos
tff(fact_2924_one__add__one,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2925_one__add__one,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2926_one__add__one,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2927_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2928_one__add__one,axiom,
    aa_int_int(plus_plus_int(one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_2929_power__strict__decreasing__iff,axiom,
    ! [B2: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(B2,M)),power_8256067586552552935nteger(B2,Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2930_power__strict__decreasing__iff,axiom,
    ! [B2: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2931_power__strict__decreasing__iff,axiom,
    ! [B2: rat,M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B2),M)),aa_nat_rat(power_power_rat(B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2932_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2933_power__strict__decreasing__iff,axiom,
    ! [B2: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),M)),aa_nat_int(power_power_int(B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2934_power__increasing__iff,axiom,
    ! [B2: code_integer,Xc: nat,Ya: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(B2,Xc)),power_8256067586552552935nteger(B2,Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya) ) ) ).

% power_increasing_iff
tff(fact_2935_power__increasing__iff,axiom,
    ! [B2: real,Xc: nat,Ya: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Xc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya) ) ) ).

% power_increasing_iff
tff(fact_2936_power__increasing__iff,axiom,
    ! [B2: rat,Xc: nat,Ya: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B2),Xc)),aa_nat_rat(power_power_rat(B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya) ) ) ).

% power_increasing_iff
tff(fact_2937_power__increasing__iff,axiom,
    ! [B2: nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya) ) ) ).

% power_increasing_iff
tff(fact_2938_power__increasing__iff,axiom,
    ! [B2: int,Xc: nat,Ya: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),Xc)),aa_nat_int(power_power_int(B2),Ya))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya) ) ) ).

% power_increasing_iff
tff(fact_2939_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_2940_numeral__plus__one,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_2941_numeral__plus__one,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_2942_numeral__plus__one,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_2943_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_2944_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_2945_one__plus__numeral,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_2946_one__plus__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_2947_one__plus__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_2948_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(plus_plus_int(one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_2949_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2950_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2951_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2952_numeral__le__one__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),one) ) ).

% numeral_le_one_iff
tff(fact_2953_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_2954_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_2955_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_2956_one__less__numeral__iff,axiom,
    ! [Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,one),Nb) ) ).

% one_less_numeral_iff
tff(fact_2957_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_2958_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_2959_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_2960_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_2961_power__decreasing__iff,axiom,
    ! [B2: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),one_one_Code_integer)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(B2,M)),power_8256067586552552935nteger(B2,Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2962_power__decreasing__iff,axiom,
    ! [B2: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2963_power__decreasing__iff,axiom,
    ! [B2: rat,M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B2),M)),aa_nat_rat(power_power_rat(B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2964_power__decreasing__iff,axiom,
    ! [B2: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),one_one_nat)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2965_power__decreasing__iff,axiom,
    ! [B2: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),M)),aa_nat_int(power_power_int(B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M) ) ) ) ).

% power_decreasing_iff
tff(fact_2966_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_2967_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_2968_one__reorient,axiom,
    ! [Xc: assn] :
      ( ( one_one_assn = Xc )
    <=> ( Xc = one_one_assn ) ) ).

% one_reorient
tff(fact_2969_one__reorient,axiom,
    ! [Xc: real] :
      ( ( one_one_real = Xc )
    <=> ( Xc = one_one_real ) ) ).

% one_reorient
tff(fact_2970_one__reorient,axiom,
    ! [Xc: rat] :
      ( ( one_one_rat = Xc )
    <=> ( Xc = one_one_rat ) ) ).

% one_reorient
tff(fact_2971_one__reorient,axiom,
    ! [Xc: nat] :
      ( ( one_one_nat = Xc )
    <=> ( Xc = one_one_nat ) ) ).

% one_reorient
tff(fact_2972_one__reorient,axiom,
    ! [Xc: int] :
      ( ( one_one_int = Xc )
    <=> ( Xc = one_one_int ) ) ).

% one_reorient
tff(fact_2973_numeral__Bit1,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),numera6690914467698888265omplex(Nb))),one_one_complex) ).

% numeral_Bit1
tff(fact_2974_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Nb)),aa_num_real(numeral_numeral_real,Nb))),one_one_real) ).

% numeral_Bit1
tff(fact_2975_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb))),one_one_rat) ).

% numeral_Bit1
tff(fact_2976_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Nb))),one_one_nat) ).

% numeral_Bit1
tff(fact_2977_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_2978_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2979_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2980_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_2981_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2982_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2983_le__numeral__extra_I4_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),one_one_real) ).

% le_numeral_extra(4)
tff(fact_2984_le__numeral__extra_I4_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),one_one_rat) ).

% le_numeral_extra(4)
tff(fact_2985_le__numeral__extra_I4_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),one_one_nat) ).

% le_numeral_extra(4)
tff(fact_2986_le__numeral__extra_I4_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),one_one_int) ).

% le_numeral_extra(4)
tff(fact_2987_less__numeral__extra_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),one_one_real) ).

% less_numeral_extra(4)
tff(fact_2988_less__numeral__extra_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),one_one_rat) ).

% less_numeral_extra(4)
tff(fact_2989_less__numeral__extra_I4_J,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),one_one_nat) ).

% less_numeral_extra(4)
tff(fact_2990_less__numeral__extra_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),one_one_int) ).

% less_numeral_extra(4)
tff(fact_2991_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_2992_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(one_one_rat),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_2993_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_2994_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: int] : aa_int_int(times_times_int(one_one_int),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_2995_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(one_one_assn),A2) = A2 ).

% comm_monoid_mult_class.mult_1
tff(fact_2996_mult_Ocomm__neutral,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),one_one_real) = A2 ).

% mult.comm_neutral
tff(fact_2997_mult_Ocomm__neutral,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(A2),one_one_rat) = A2 ).

% mult.comm_neutral
tff(fact_2998_mult_Ocomm__neutral,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),one_one_nat) = A2 ).

% mult.comm_neutral
tff(fact_2999_mult_Ocomm__neutral,axiom,
    ! [A2: int] : aa_int_int(times_times_int(A2),one_one_int) = A2 ).

% mult.comm_neutral
tff(fact_3000_mult_Ocomm__neutral,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(A2),one_one_assn) = A2 ).

% mult.comm_neutral
tff(fact_3001_power__eq__if,axiom,
    ! [P3: complex,M: nat] :
      aa_nat_complex(power_power_complex(P3),M) = $ite(M = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P3),aa_nat_complex(power_power_complex(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3002_power__eq__if,axiom,
    ! [P3: code_integer,M: nat] :
      power_8256067586552552935nteger(P3,M) = $ite(M = zero_zero_nat,one_one_Code_integer,times_3573771949741848930nteger(P3,power_8256067586552552935nteger(P3,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3003_power__eq__if,axiom,
    ! [P3: real,M: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),M) = $ite(M = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P3),aa_nat_real(aa_real_fun_nat_real(power_power_real,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3004_power__eq__if,axiom,
    ! [P3: rat,M: nat] :
      aa_nat_rat(power_power_rat(P3),M) = $ite(M = zero_zero_nat,one_one_rat,aa_rat_rat(times_times_rat(P3),aa_nat_rat(power_power_rat(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3005_power__eq__if,axiom,
    ! [P3: nat,M: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),M) = $ite(M = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3006_power__eq__if,axiom,
    ! [P3: int,M: nat] :
      aa_nat_int(power_power_int(P3),M) = $ite(M = zero_zero_nat,one_one_int,aa_int_int(times_times_int(P3),aa_nat_int(power_power_int(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3007_power__eq__if,axiom,
    ! [P3: assn,M: nat] :
      aa_nat_assn(power_power_assn(P3),M) = $ite(M = zero_zero_nat,one_one_assn,aa_assn_assn(times_times_assn(P3),aa_nat_assn(power_power_assn(P3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))) ).

% power_eq_if
tff(fact_3008_num_Oexhaust,axiom,
    ! [Ya: num] :
      ( ( Ya != one )
     => ( ! [X23: num] : Ya != aa_num_num(bit0,X23)
       => ~ ! [X32: num] : Ya != aa_num_num(bit1,X32) ) ) ).

% num.exhaust
tff(fact_3009_xor__num_Ocases,axiom,
    ! [Xc: product_prod_num_num] :
      ( ( Xc != product_Pair_num_num(one,one) )
     => ( ! [N: num] : Xc != product_Pair_num_num(one,aa_num_num(bit0,N))
       => ( ! [N: num] : Xc != product_Pair_num_num(one,aa_num_num(bit1,N))
         => ( ! [M4: num] : Xc != product_Pair_num_num(aa_num_num(bit0,M4),one)
           => ( ! [M4: num,N: num] : Xc != product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit0,N))
             => ( ! [M4: num,N: num] : Xc != product_Pair_num_num(aa_num_num(bit0,M4),aa_num_num(bit1,N))
               => ( ! [M4: num] : Xc != product_Pair_num_num(aa_num_num(bit1,M4),one)
                 => ( ! [M4: num,N: num] : Xc != product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit0,N))
                   => ~ ! [M4: num,N: num] : Xc != product_Pair_num_num(aa_num_num(bit1,M4),aa_num_num(bit1,N)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_3010_discrete,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),B2) ) ).

% discrete
tff(fact_3011_discrete,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(A2),one_one_int)),B2) ) ).

% discrete
tff(fact_3012_not__one__le__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),zero_zero_real) ).

% not_one_le_zero
tff(fact_3013_not__one__le__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),zero_zero_rat) ).

% not_one_le_zero
tff(fact_3014_not__one__le__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),zero_zero_nat) ).

% not_one_le_zero
tff(fact_3015_not__one__le__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),zero_zero_int) ).

% not_one_le_zero
tff(fact_3016_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_3017_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_3018_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_3019_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_3020_zero__less__one__class_Ozero__le__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),one_one_real) ).

% zero_less_one_class.zero_le_one
tff(fact_3021_zero__less__one__class_Ozero__le__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one_class.zero_le_one
tff(fact_3022_zero__less__one__class_Ozero__le__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one_class.zero_le_one
tff(fact_3023_zero__less__one__class_Ozero__le__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),one_one_int) ).

% zero_less_one_class.zero_le_one
tff(fact_3024_less__numeral__extra_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% less_numeral_extra(1)
tff(fact_3025_less__numeral__extra_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% less_numeral_extra(1)
tff(fact_3026_less__numeral__extra_I1_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% less_numeral_extra(1)
tff(fact_3027_less__numeral__extra_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% less_numeral_extra(1)
tff(fact_3028_zero__less__one,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),one_one_real) ).

% zero_less_one
tff(fact_3029_zero__less__one,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),one_one_rat) ).

% zero_less_one
tff(fact_3030_zero__less__one,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),one_one_nat) ).

% zero_less_one
tff(fact_3031_zero__less__one,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),one_one_int) ).

% zero_less_one
tff(fact_3032_not__one__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),zero_zero_real) ).

% not_one_less_zero
tff(fact_3033_not__one__less__zero,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),zero_zero_rat) ).

% not_one_less_zero
tff(fact_3034_not__one__less__zero,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),zero_zero_nat) ).

% not_one_less_zero
tff(fact_3035_not__one__less__zero,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),zero_zero_int) ).

% not_one_less_zero
tff(fact_3036_one__le__numeral,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_num_real(numeral_numeral_real,Nb)) ).

% one_le_numeral
tff(fact_3037_one__le__numeral,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_le_numeral
tff(fact_3038_one__le__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% one_le_numeral
tff(fact_3039_one__le__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) ).

% one_le_numeral
tff(fact_3040_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),one_one_real) ).

% not_numeral_less_one
tff(fact_3041_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) ).

% not_numeral_less_one
tff(fact_3042_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Nb)),one_one_nat) ).

% not_numeral_less_one
tff(fact_3043_not__numeral__less__one,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) ).

% not_numeral_less_one
tff(fact_3044_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_3045_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_3046_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_3047_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_3048_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_3049_one__plus__numeral__commute,axiom,
    ! [Xc: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Xc)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xc)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_3050_one__plus__numeral__commute,axiom,
    ! [Xc: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xc)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_3051_one__plus__numeral__commute,axiom,
    ! [Xc: num] : aa_rat_rat(plus_plus_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,Xc)) = aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,Xc)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_3052_one__plus__numeral__commute,axiom,
    ! [Xc: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xc)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xc)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_3053_one__plus__numeral__commute,axiom,
    ! [Xc: num] : aa_int_int(plus_plus_int(one_one_int),aa_num_int(numeral_numeral_int,Xc)) = aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,Xc)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_3054_less__add__one,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)) ).

% less_add_one
tff(fact_3055_less__add__one,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(plus_plus_rat(A2),one_one_rat)) ).

% less_add_one
tff(fact_3056_less__add__one,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)) ).

% less_add_one
tff(fact_3057_less__add__one,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(plus_plus_int(A2),one_one_int)) ).

% less_add_one
tff(fact_3058_add__mono1,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),one_one_real)) ) ).

% add_mono1
tff(fact_3059_add__mono1,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(A2),one_one_rat)),aa_rat_rat(plus_plus_rat(B2),one_one_rat)) ) ).

% add_mono1
tff(fact_3060_add__mono1,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),one_one_nat)) ) ).

% add_mono1
tff(fact_3061_add__mono1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),one_one_int)),aa_int_int(plus_plus_int(B2),one_one_int)) ) ).

% add_mono1
tff(fact_3062_less__1__mult,axiom,
    ! [M: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),Nb)) ) ) ).

% less_1_mult
tff(fact_3063_less__1__mult,axiom,
    ! [M: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),M)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(M),Nb)) ) ) ).

% less_1_mult
tff(fact_3064_less__1__mult,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) ) ) ).

% less_1_mult
tff(fact_3065_less__1__mult,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(times_times_int(M),Nb)) ) ) ).

% less_1_mult
tff(fact_3066_right__inverse__eq,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A2),B2) = one_one_complex )
      <=> ( A2 = B2 ) ) ) ).

% right_inverse_eq
tff(fact_3067_right__inverse__eq,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) = one_one_real )
      <=> ( A2 = B2 ) ) ) ).

% right_inverse_eq
tff(fact_3068_right__inverse__eq,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A2),B2) = one_one_rat )
      <=> ( A2 = B2 ) ) ) ).

% right_inverse_eq
tff(fact_3069_one__le__power,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ).

% one_le_power
tff(fact_3070_one__le__power,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),power_8256067586552552935nteger(A2,Nb)) ) ).

% one_le_power
tff(fact_3071_one__le__power,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),Nb)) ) ).

% one_le_power
tff(fact_3072_one__le__power,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ) ).

% one_le_power
tff(fact_3073_one__le__power,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A2),Nb)) ) ).

% one_le_power
tff(fact_3074_left__right__inverse__power,axiom,
    ! [Xc: complex,Ya: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xc),Ya) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xc),Nb)),aa_nat_complex(power_power_complex(Ya),Nb)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_3075_left__right__inverse__power,axiom,
    ! [Xc: code_integer,Ya: code_integer,Nb: nat] :
      ( ( times_3573771949741848930nteger(Xc,Ya) = one_one_Code_integer )
     => ( times_3573771949741848930nteger(power_8256067586552552935nteger(Xc,Nb),power_8256067586552552935nteger(Ya,Nb)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_3076_left__right__inverse__power,axiom,
    ! [Xc: real,Ya: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),Nb)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_3077_left__right__inverse__power,axiom,
    ! [Xc: rat,Ya: rat,Nb: nat] :
      ( ( aa_rat_rat(times_times_rat(Xc),Ya) = one_one_rat )
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(Xc),Nb)),aa_nat_rat(power_power_rat(Ya),Nb)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_3078_left__right__inverse__power,axiom,
    ! [Xc: nat,Ya: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xc),Ya) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ya),Nb)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_3079_left__right__inverse__power,axiom,
    ! [Xc: int,Ya: int,Nb: nat] :
      ( ( aa_int_int(times_times_int(Xc),Ya) = one_one_int )
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(Xc),Nb)),aa_nat_int(power_power_int(Ya),Nb)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_3080_left__right__inverse__power,axiom,
    ! [Xc: assn,Ya: assn,Nb: nat] :
      ( ( aa_assn_assn(times_times_assn(Xc),Ya) = one_one_assn )
     => ( aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(Xc),Nb)),aa_nat_assn(power_power_assn(Ya),Nb)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_3081_power__one__over,axiom,
    ! [A2: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A2)),Nb) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_nat_complex(power_power_complex(A2),Nb)) ).

% power_one_over
tff(fact_3082_power__one__over,axiom,
    ! [A2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A2)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ).

% power_one_over
tff(fact_3083_power__one__over,axiom,
    ! [A2: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(one_one_rat),A2)),Nb) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(power_power_rat(A2),Nb)) ).

% power_one_over
tff(fact_3084_power__0,axiom,
    ! [A2: assn] : aa_nat_assn(power_power_assn(A2),zero_zero_nat) = one_one_assn ).

% power_0
tff(fact_3085_power__0,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(A2),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_3086_power__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_3087_power__0,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),zero_zero_nat) = one_one_real ).

% power_0
tff(fact_3088_power__0,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_3089_power__0,axiom,
    ! [A2: complex] : aa_nat_complex(power_power_complex(A2),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_3090_power__0,axiom,
    ! [A2: code_integer] : power_8256067586552552935nteger(A2,zero_zero_nat) = one_one_Code_integer ).

% power_0
tff(fact_3091_real__arch__pow,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),N)) ) ).

% real_arch_pow
tff(fact_3092_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% eval_nat_numeral(3)
tff(fact_3093_mult__left__le__one__le,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Xc)),Xc) ) ) ) ).

% mult_left_le_one_le
tff(fact_3094_mult__left__le__one__le,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Ya),Xc)),Xc) ) ) ) ).

% mult_left_le_one_le
tff(fact_3095_mult__left__le__one__le,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Ya),Xc)),Xc) ) ) ) ).

% mult_left_le_one_le
tff(fact_3096_mult__right__le__one__le,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya)),Xc) ) ) ) ).

% mult_right_le_one_le
tff(fact_3097_mult__right__le__one__le,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xc),Ya)),Xc) ) ) ) ).

% mult_right_le_one_le
tff(fact_3098_mult__right__le__one__le,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Xc),Ya)),Xc) ) ) ) ).

% mult_right_le_one_le
tff(fact_3099_mult__le__one,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_3100_mult__le__one,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),B2)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_3101_mult__le__one,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_3102_mult__le__one,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),B2)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_3103_mult__left__le,axiom,
    ! [C2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),A2) ) ) ).

% mult_left_le
tff(fact_3104_mult__left__le,axiom,
    ! [C2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),A2) ) ) ).

% mult_left_le
tff(fact_3105_mult__left__le,axiom,
    ! [C2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),A2) ) ) ).

% mult_left_le
tff(fact_3106_mult__left__le,axiom,
    ! [C2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),C2)),A2) ) ) ).

% mult_left_le
tff(fact_3107_zero__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real)) ).

% zero_less_two
tff(fact_3108_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_3109_zero__less__two,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat)) ).

% zero_less_two
tff(fact_3110_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_3111_divide__less__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_3112_divide__less__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_3113_less__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ) ) ).

% less_divide_eq_1
tff(fact_3114_less__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ) ) ).

% less_divide_eq_1
tff(fact_3115_power__le__one,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),one_one_real) ) ) ).

% power_le_one
tff(fact_3116_power__le__one,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,Nb)),one_one_Code_integer) ) ) ).

% power_le_one
tff(fact_3117_power__le__one,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),Nb)),one_one_rat) ) ) ).

% power_le_one
tff(fact_3118_power__le__one,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),one_one_nat) ) ) ).

% power_le_one
tff(fact_3119_power__le__one,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),Nb)),one_one_int) ) ) ).

% power_le_one
tff(fact_3120_div__add__self1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_3121_div__add__self1,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(B2),A2)),B2) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),one_one_int) ) ) ).

% div_add_self1
tff(fact_3122_div__add__self2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_3123_div__add__self2,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A2),B2)),B2) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),one_one_int) ) ) ).

% div_add_self2
tff(fact_3124_gt__half__sum,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B2) ) ).

% gt_half_sum
tff(fact_3125_gt__half__sum,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A2),B2)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))),B2) ) ).

% gt_half_sum
tff(fact_3126_less__half__sum,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_3127_less__half__sum,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(A2),B2)),aa_rat_rat(plus_plus_rat(one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_3128_square__diff__one__factored,axiom,
    ! [Xc: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),one_one_real)) ).

% square_diff_one_factored
tff(fact_3129_square__diff__one__factored,axiom,
    ! [Xc: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(times_times_rat(Xc),Xc)),one_one_rat) = aa_rat_rat(times_times_rat(aa_rat_rat(plus_plus_rat(Xc),one_one_rat)),aa_rat_rat(minus_minus_rat(Xc),one_one_rat)) ).

% square_diff_one_factored
tff(fact_3130_square__diff__one__factored,axiom,
    ! [Xc: int] : aa_int_int(minus_minus_int(aa_int_int(times_times_int(Xc),Xc)),one_one_int) = aa_int_int(times_times_int(aa_int_int(plus_plus_int(Xc),one_one_int)),aa_int_int(minus_minus_int(Xc),one_one_int)) ).

% square_diff_one_factored
tff(fact_3131_power__gt1__lemma,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),times_3573771949741848930nteger(A2,power_8256067586552552935nteger(A2,Nb))) ) ).

% power_gt1_lemma
tff(fact_3132_power__gt1__lemma,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb))) ) ).

% power_gt1_lemma
tff(fact_3133_power__gt1__lemma,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),Nb))) ) ).

% power_gt1_lemma
tff(fact_3134_power__gt1__lemma,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb))) ) ).

% power_gt1_lemma
tff(fact_3135_power__gt1__lemma,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(times_times_int(A2),aa_nat_int(power_power_int(A2),Nb))) ) ).

% power_gt1_lemma
tff(fact_3136_power__less__power__Suc,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(A2,Nb)),times_3573771949741848930nteger(A2,power_8256067586552552935nteger(A2,Nb))) ) ).

% power_less_power_Suc
tff(fact_3137_power__less__power__Suc,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb))) ) ).

% power_less_power_Suc
tff(fact_3138_power__less__power__Suc,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),Nb)),aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),Nb))) ) ).

% power_less_power_Suc
tff(fact_3139_power__less__power__Suc,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb))) ) ).

% power_less_power_Suc
tff(fact_3140_power__less__power__Suc,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),Nb)),aa_int_int(times_times_int(A2),aa_nat_int(power_power_int(A2),Nb))) ) ).

% power_less_power_Suc
tff(fact_3141_power__gt1,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),power_8256067586552552935nteger(A2,aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_3142_power__gt1,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_3143_power__gt1,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_3144_power__gt1,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_3145_power__gt1,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A2),aa_nat_nat(suc,Nb))) ) ).

% power_gt1
tff(fact_3146_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_3147_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% power_0_left
tff(fact_3148_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% power_0_left
tff(fact_3149_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% power_0_left
tff(fact_3150_power__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% power_0_left
tff(fact_3151_power__0__left,axiom,
    ! [Nb: nat] :
      power_8256067586552552935nteger(zero_z3403309356797280102nteger,Nb) = $ite(Nb = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ).

% power_0_left
tff(fact_3152_power__less__imp__less__exp,axiom,
    ! [A2: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(A2,M)),power_8256067586552552935nteger(A2,Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_3153_power__less__imp__less__exp,axiom,
    ! [A2: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_3154_power__less__imp__less__exp,axiom,
    ! [A2: rat,M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_3155_power__less__imp__less__exp,axiom,
    ! [A2: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_3156_power__less__imp__less__exp,axiom,
    ! [A2: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_3157_power__strict__increasing,axiom,
    ! [Nb: nat,N5: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(A2,Nb)),power_8256067586552552935nteger(A2,N5)) ) ) ).

% power_strict_increasing
tff(fact_3158_power__strict__increasing,axiom,
    ! [Nb: nat,N5: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N5)) ) ) ).

% power_strict_increasing
tff(fact_3159_power__strict__increasing,axiom,
    ! [Nb: nat,N5: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),Nb)),aa_nat_rat(power_power_rat(A2),N5)) ) ) ).

% power_strict_increasing
tff(fact_3160_power__strict__increasing,axiom,
    ! [Nb: nat,N5: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N5)) ) ) ).

% power_strict_increasing
tff(fact_3161_power__strict__increasing,axiom,
    ! [Nb: nat,N5: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(A2),N5)) ) ) ).

% power_strict_increasing
tff(fact_3162_power__increasing,axiom,
    ! [Nb: nat,N5: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N5)) ) ) ).

% power_increasing
tff(fact_3163_power__increasing,axiom,
    ! [Nb: nat,N5: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,Nb)),power_8256067586552552935nteger(A2,N5)) ) ) ).

% power_increasing
tff(fact_3164_power__increasing,axiom,
    ! [Nb: nat,N5: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),Nb)),aa_nat_rat(power_power_rat(A2),N5)) ) ) ).

% power_increasing
tff(fact_3165_power__increasing,axiom,
    ! [Nb: nat,N5: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N5)) ) ) ).

% power_increasing
tff(fact_3166_power__increasing,axiom,
    ! [Nb: nat,N5: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(A2),N5)) ) ) ).

% power_increasing
tff(fact_3167_real__arch__pow__inv,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real)
       => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),N)),Ya) ) ) ).

% real_arch_pow_inv
tff(fact_3168_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,Nb) ).

% numeral_Bit1_div_2
tff(fact_3169_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,Nb) ).

% numeral_Bit1_div_2
tff(fact_3170_power3__eq__cube,axiom,
    ! [A2: complex] : aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3171_power3__eq__cube,axiom,
    ! [A2: code_integer] : power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = times_3573771949741848930nteger(times_3573771949741848930nteger(A2,A2),A2) ).

% power3_eq_cube
tff(fact_3172_power3__eq__cube,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3173_power3__eq__cube,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3174_power3__eq__cube,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3175_power3__eq__cube,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(times_times_int(aa_int_int(times_times_int(A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3176_power3__eq__cube,axiom,
    ! [A2: assn] : aa_nat_assn(power_power_assn(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),A2)),A2) ).

% power3_eq_cube
tff(fact_3177_numeral__3__eq__3,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) = aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))) ).

% numeral_3_eq_3
tff(fact_3178_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb) ).

% Suc3_eq_add_3
tff(fact_3179_mult__le__cancel__left1,axiom,
    ! [C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3180_mult__le__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(times_times_rat(C2),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3181_mult__le__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(times_times_int(C2),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_3182_mult__le__cancel__left2,axiom,
    ! [C2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A2) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3183_mult__le__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(C2),A2)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3184_mult__le__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(C2),A2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2) ) ) ) ).

% mult_le_cancel_left2
tff(fact_3185_mult__le__cancel__right1,axiom,
    ! [C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3186_mult__le__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(times_times_rat(B2),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3187_mult__le__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(times_times_int(B2),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_3188_mult__le__cancel__right2,axiom,
    ! [A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A2) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3189_mult__le__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3190_mult__le__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(A2),C2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2) ) ) ) ).

% mult_le_cancel_right2
tff(fact_3191_mult__less__cancel__left1,axiom,
    ! [C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3192_mult__less__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(times_times_rat(C2),B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3193_mult__less__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(times_times_int(C2),B2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_3194_mult__less__cancel__left2,axiom,
    ! [C2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3195_mult__less__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(C2),A2)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3196_mult__less__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(C2),A2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2) ) ) ) ).

% mult_less_cancel_left2
tff(fact_3197_mult__less__cancel__right1,axiom,
    ! [C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3198_mult__less__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(times_times_rat(B2),C2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B2) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3199_mult__less__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(times_times_int(B2),C2))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B2) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_3200_mult__less__cancel__right2,axiom,
    ! [A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),C2)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),one_one_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3201_mult__less__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),C2)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3202_mult__less__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),C2)),C2)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2) ) ) ) ).

% mult_less_cancel_right2
tff(fact_3203_field__le__mult__one__interval,axiom,
    ! [Xc: real,Ya: real] :
      ( ! [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z2)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z2),Xc)),Ya) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ).

% field_le_mult_one_interval
tff(fact_3204_field__le__mult__one__interval,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ! [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z2),Xc)),Ya) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya) ) ).

% field_le_mult_one_interval
tff(fact_3205_convex__bound__le,axiom,
    ! [Xc: real,A2: real,Ya: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Ya))),A2) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3206_convex__bound__le,axiom,
    ! [Xc: rat,A2: rat,Ya: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xc)),aa_rat_rat(times_times_rat(V),Ya))),A2) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3207_convex__bound__le,axiom,
    ! [Xc: int,A2: int,Ya: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(plus_plus_int(U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(U),Xc)),aa_int_int(times_times_int(V),Ya))),A2) ) ) ) ) ) ).

% convex_bound_le
tff(fact_3208_divide__le__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2)),one_one_real)
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_3209_divide__le__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),A2)),one_one_rat)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_3210_le__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) )
        | ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ) ).

% le_divide_eq_1
tff(fact_3211_le__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),A2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) )
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ) ).

% le_divide_eq_1
tff(fact_3212_power__Suc__less,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(A2,power_8256067586552552935nteger(A2,Nb))),power_8256067586552552935nteger(A2,Nb)) ) ) ).

% power_Suc_less
tff(fact_3213_power__Suc__less,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ) ).

% power_Suc_less
tff(fact_3214_power__Suc__less,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),aa_nat_rat(power_power_rat(A2),Nb))),aa_nat_rat(power_power_rat(A2),Nb)) ) ) ).

% power_Suc_less
tff(fact_3215_power__Suc__less,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ) ) ).

% power_Suc_less
tff(fact_3216_power__Suc__less,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(A2),aa_nat_int(power_power_int(A2),Nb))),aa_nat_int(power_power_int(A2),Nb)) ) ) ).

% power_Suc_less
tff(fact_3217_power__Suc__le__self,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,Nb))),A2) ) ) ).

% power_Suc_le_self
tff(fact_3218_power__Suc__le__self,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,aa_nat_nat(suc,Nb))),A2) ) ) ).

% power_Suc_le_self
tff(fact_3219_power__Suc__le__self,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,Nb))),A2) ) ) ).

% power_Suc_le_self
tff(fact_3220_power__Suc__le__self,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,Nb))),A2) ) ) ).

% power_Suc_le_self
tff(fact_3221_power__Suc__le__self,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,Nb))),A2) ) ) ).

% power_Suc_le_self
tff(fact_3222_power__Suc__less__one,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(A2,aa_nat_nat(suc,Nb))),one_one_Code_integer) ) ) ).

% power_Suc_less_one
tff(fact_3223_power__Suc__less__one,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(suc,Nb))),one_one_real) ) ) ).

% power_Suc_less_one
tff(fact_3224_power__Suc__less__one,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_nat_nat(suc,Nb))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_3225_power__Suc__less__one,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(suc,Nb))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_3226_power__Suc__less__one,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),aa_nat_nat(suc,Nb))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_3227_power__strict__decreasing,axiom,
    ! [Nb: nat,N5: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(A2,N5)),power_8256067586552552935nteger(A2,Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3228_power__strict__decreasing,axiom,
    ! [Nb: nat,N5: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3229_power__strict__decreasing,axiom,
    ! [Nb: nat,N5: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),N5)),aa_nat_rat(power_power_rat(A2),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3230_power__strict__decreasing,axiom,
    ! [Nb: nat,N5: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N5)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3231_power__strict__decreasing,axiom,
    ! [Nb: nat,N5: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),N5)),aa_nat_int(power_power_int(A2),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_3232_one__power2,axiom,
    aa_nat_rat(power_power_rat(one_one_rat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat ).

% one_power2
tff(fact_3233_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_power2
tff(fact_3234_one__power2,axiom,
    aa_nat_real(aa_real_fun_nat_real(power_power_real,one_one_real),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real ).

% one_power2
tff(fact_3235_one__power2,axiom,
    aa_nat_int(power_power_int(one_one_int),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int ).

% one_power2
tff(fact_3236_one__power2,axiom,
    aa_nat_complex(power_power_complex(one_one_complex),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex ).

% one_power2
tff(fact_3237_one__power2,axiom,
    power_8256067586552552935nteger(one_one_Code_integer,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_power2
tff(fact_3238_power__decreasing,axiom,
    ! [Nb: nat,N5: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),N5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3239_power__decreasing,axiom,
    ! [Nb: nat,N5: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),one_one_Code_integer)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,N5)),power_8256067586552552935nteger(A2,Nb)) ) ) ) ).

% power_decreasing
tff(fact_3240_power__decreasing,axiom,
    ! [Nb: nat,N5: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),N5)),aa_nat_rat(power_power_rat(A2),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3241_power__decreasing,axiom,
    ! [Nb: nat,N5: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),one_one_nat)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),N5)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3242_power__decreasing,axiom,
    ! [Nb: nat,N5: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),N5)),aa_nat_int(power_power_int(A2),Nb)) ) ) ) ).

% power_decreasing
tff(fact_3243_power__le__imp__le__exp,axiom,
    ! [A2: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,M)),power_8256067586552552935nteger(A2,Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_3244_power__le__imp__le__exp,axiom,
    ! [A2: real,M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_3245_power__le__imp__le__exp,axiom,
    ! [A2: rat,M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),M)),aa_nat_rat(power_power_rat(A2),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_3246_power__le__imp__le__exp,axiom,
    ! [A2: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_3247_power__le__imp__le__exp,axiom,
    ! [A2: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_3248_self__le__power,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ) ).

% self_le_power
tff(fact_3249_self__le__power,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),power_8256067586552552935nteger(A2,Nb)) ) ) ).

% self_le_power
tff(fact_3250_self__le__power,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_nat_rat(power_power_rat(A2),Nb)) ) ) ).

% self_le_power
tff(fact_3251_self__le__power,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ) ) ).

% self_le_power
tff(fact_3252_self__le__power,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(A2),Nb)) ) ) ).

% self_le_power
tff(fact_3253_one__less__power,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),power_8256067586552552935nteger(A2,Nb)) ) ) ).

% one_less_power
tff(fact_3254_one__less__power,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ) ).

% one_less_power
tff(fact_3255_one__less__power,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_nat_rat(power_power_rat(A2),Nb)) ) ) ).

% one_less_power
tff(fact_3256_one__less__power,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ) ) ).

% one_less_power
tff(fact_3257_one__less__power,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A2),Nb)) ) ) ).

% one_less_power
tff(fact_3258_num_Osize_I6_J,axiom,
    ! [X33: num] : aa_num_nat(size_size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size(6)
tff(fact_3259_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_3260_convex__bound__lt,axiom,
    ! [Xc: real,A2: real,Ya: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Ya))),A2) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3261_convex__bound__lt,axiom,
    ! [Xc: rat,A2: rat,Ya: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(plus_plus_rat(U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(U),Xc)),aa_rat_rat(times_times_rat(V),Ya))),A2) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3262_convex__bound__lt,axiom,
    ! [Xc: int,A2: int,Ya: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),A2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(plus_plus_int(U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(U),Xc)),aa_int_int(times_times_int(V),Ya))),A2) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3263_power__diff__power__eq,axiom,
    ! [A2: code_integer,M: nat,Nb: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(power_8256067586552552935nteger(A2,M),power_8256067586552552935nteger(A2,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),power_8256067586552552935nteger(A2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),divide6298287555418463151nteger(one_one_Code_integer,power_8256067586552552935nteger(A2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3264_power__diff__power__eq,axiom,
    ! [A2: nat,M: nat,Nb: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3265_power__diff__power__eq,axiom,
    ! [A2: int,M: nat,Nb: nat] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M),aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ).

% power_diff_power_eq
tff(fact_3266_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% two_realpow_ge_one
tff(fact_3267_combine__options__cases,axiom,
    ! [Xc: option_nat,P: fun_op5794009454290973322_nat_o,Ya: option_nat] :
      ( ( ( Xc = none_nat )
       => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xc),Ya) )
     => ( ( ( Ya = none_nat )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xc),Ya) )
       => ( ! [A3: nat,B3: nat] :
              ( ( Xc = some_nat(A3) )
             => ( ( Ya = some_nat(B3) )
               => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xc),Ya) ) )
         => aa_option_nat_o(aa_opt4319133605541840243_nat_o(P,Xc),Ya) ) ) ) ).

% combine_options_cases
tff(fact_3268_combine__options__cases,axiom,
    ! [Xc: option_nat,P: fun_op3832135881594366065_nat_o,Ya: option4927543243414619207at_nat] :
      ( ( ( Xc = none_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xc),Ya) )
     => ( ( ( Ya = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xc),Ya) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( Xc = some_nat(A3) )
             => ( ( Ya = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xc),Ya) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt120907569613471666_nat_o(P,Xc),Ya) ) ) ) ).

% combine_options_cases
tff(fact_3269_combine__options__cases,axiom,
    ! [Xc: option4927543243414619207at_nat,P: fun_op7326680082016885773_nat_o,Ya: option_nat] :
      ( ( ( Xc = none_P5556105721700978146at_nat )
       => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xc),Ya) )
     => ( ( ( Ya = none_nat )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xc),Ya) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( Xc = some_P7363390416028606310at_nat(A3) )
             => ( ( Ya = some_nat(B3) )
               => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xc),Ya) ) )
         => aa_option_nat_o(aa_opt7110402928079307526_nat_o(P,Xc),Ya) ) ) ) ).

% combine_options_cases
tff(fact_3270_combine__options__cases,axiom,
    ! [Xc: option4927543243414619207at_nat,P: fun_op1221328669798608686_nat_o,Ya: option4927543243414619207at_nat] :
      ( ( ( Xc = none_P5556105721700978146at_nat )
       => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xc),Ya) )
     => ( ( ( Ya = none_P5556105721700978146at_nat )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xc),Ya) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( Xc = some_P7363390416028606310at_nat(A3) )
             => ( ( Ya = some_P7363390416028606310at_nat(B3) )
               => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xc),Ya) ) )
         => aa_opt7112393969400857165_nat_o(aa_opt9039718665754746463_nat_o(P,Xc),Ya) ) ) ) ).

% combine_options_cases
tff(fact_3271_split__option__all,axiom,
    ! [P: fun_option_nat_o] :
      ( ! [X_1: option_nat] : aa_option_nat_o(P,X_1)
    <=> ( aa_option_nat_o(P,none_nat)
        & ! [X2: nat] : aa_option_nat_o(P,some_nat(X2)) ) ) ).

% split_option_all
tff(fact_3272_split__option__all,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ! [X_1: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_1)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        & ! [X2: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X2)) ) ) ).

% split_option_all
tff(fact_3273_split__option__ex,axiom,
    ! [P: fun_option_nat_o] :
      ( ? [X_1: option_nat] : aa_option_nat_o(P,X_1)
    <=> ( aa_option_nat_o(P,none_nat)
        | ? [X2: nat] : aa_option_nat_o(P,some_nat(X2)) ) ) ).

% split_option_ex
tff(fact_3274_split__option__ex,axiom,
    ! [P: fun_op8028897214545316308_nat_o] :
      ( ? [X_1: option4927543243414619207at_nat] : aa_opt7112393969400857165_nat_o(P,X_1)
    <=> ( aa_opt7112393969400857165_nat_o(P,none_P5556105721700978146at_nat)
        | ? [X2: product_prod_nat_nat] : aa_opt7112393969400857165_nat_o(P,some_P7363390416028606310at_nat(X2)) ) ) ).

% split_option_ex
tff(fact_3275_option_Oexhaust,axiom,
    ! [Ya: option_nat] :
      ( ( Ya != none_nat )
     => ~ ! [X23: nat] : Ya != some_nat(X23) ) ).

% option.exhaust
tff(fact_3276_option_Oexhaust,axiom,
    ! [Ya: option4927543243414619207at_nat] :
      ( ( Ya != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] : Ya != some_P7363390416028606310at_nat(X23) ) ).

% option.exhaust
tff(fact_3277_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option = some_nat(X22) )
     => ( Option != none_nat ) ) ).

% option.discI
tff(fact_3278_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option = some_P7363390416028606310at_nat(X22) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
tff(fact_3279_option_Odistinct_I1_J,axiom,
    ! [X22: nat] : none_nat != some_nat(X22) ).

% option.distinct(1)
tff(fact_3280_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] : none_P5556105721700978146at_nat != some_P7363390416028606310at_nat(X22) ).

% option.distinct(1)
tff(fact_3281_option_Osel,axiom,
    ! [X22: nat] : the_nat(some_nat(X22)) = X22 ).

% option.sel
tff(fact_3282_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] : the_Pr8591224930841456533at_nat(some_P7363390416028606310at_nat(X22)) = X22 ).

% option.sel
tff(fact_3283_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
      <=> ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( the_nat(Option) = the_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_3284_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
      <=> ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( the_Pr8591224930841456533at_nat(Option) = the_Pr8591224930841456533at_nat(Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_3285_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I3,A,Xs,Xsi)),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_3286_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,I: nat,Xsi: list_nat,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L8650695023172932196BT_nat(I3,A,Xs,Xsi)),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_3287_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_re8050108611700907676T_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I3,A,Xs,Xsi)),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_3288_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_re560840401387337987i_assn,Xs: list_real,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I3,A,Xs),Xsi)),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_3289_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_re574865750389657758t_assn,Xs: list_real,I: nat,Xsi: list_nat,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L234762979517870878al_nat(I3,A,Xs,Xsi)),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_3290_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_o_4353168661238401050T_assn,Xs: list_o,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L1319876754960170684T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L1319876754960170684T_VEBT(I3,A,Xs,Xsi)),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_3291_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_o_9029690514724572677i_assn,Xs: list_o,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A,aa_nat_o(nth_o(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(I3,A,Xs),Xsi)),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_3292_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_o_fun_nat_assn,Xs: list_o,I: nat,Xsi: list_nat,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_o_fun_nat_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2281750874075065672_o_nat(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L2281750874075065672_o_nat(I3,A,Xs,Xsi)),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_3293_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_na4489259242115493880T_assn,Xs: list_nat,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L8511957252848910786T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L8511957252848910786T_VEBT(I3,A,Xs,Xsi)),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_3294_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: fun_na6467670216208870567i_assn,Xs: list_nat,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A,aa_nat_nat(nth_nat(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(I3,A,Xs),Xsi)),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_3295_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_VE1571388752190420990T_assn,Xc: vEBT_VEBT,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,Xc),Xi)),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I3,A,list_u1324408373059187874T_VEBT(Xs,I,Xc),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_3296_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_re560840401387337987i_assn,Xc: real,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,Xc),Xi)),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(I3,A,list_update_real(Xs,I,Xc)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_3297_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_re8050108611700907676T_assn,Xc: real,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs: list_real,Xsi: list_VEBT_VEBT,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,Xc),Xi)),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L3095048238742455910T_VEBT(I3,A,list_update_real(Xs,I,Xc),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_3298_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_o_9029690514724572677i_assn,Xc: $o,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A,(Xc)),Xi)),aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(I3,A,list_update_o(Xs,I,(Xc))),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_3299_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_o_4353168661238401050T_assn,Xc: $o,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs: list_o,Xsi: list_VEBT_VEBT,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A,(Xc)),Xi)),vEBT_L1319876754960170684T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L1319876754960170684T_VEBT(I3,A,list_update_o(Xs,I,(Xc)),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_3300_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_na6467670216208870567i_assn,Xc: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A,Xc),Xi)),aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(I3,A,list_update_nat(Xs,I,Xc)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_3301_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_na4489259242115493880T_assn,Xc: nat,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs: list_nat,Xsi: list_VEBT_VEBT,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A,Xc),Xi)),vEBT_L8511957252848910786T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L8511957252848910786T_VEBT(I3,A,list_update_nat(Xs,I,Xc),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_3302_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_VE2561185598974572172i_assn,Xc: vEBT_VEBTi,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(A,Xc),Xi)),aa_lis2508312872377092907i_assn(vEBT_L886525131989349516_VEBTi(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L886525131989349516_VEBTi(I3,A,list_u6098035379799741383_VEBTi(Xs,I,Xc)),list_u6098035379799741383_VEBTi(Xsi,I,Xi))),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_3303_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_VE5577048379444288083T_assn,Xc: vEBT_VEBTi,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB2970430004736285756T_assn(A,Xc),Xi)),vEBT_L2497118539674116125T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(vEBT_L2497118539674116125T_VEBT(I3,A,list_u6098035379799741383_VEBTi(Xs,I,Xc),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F),C2,Q)
           => hoare_hoare_triple_o(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_3304_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: fun_VE1571388752190420990T_assn,Xc: vEBT_VEBT,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,Xc),Xi)),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(I3,insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( aa_set_nat_o(member_nat(I),I3)
         => ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(vEBT_L3204528365124325536T_VEBT(I3,A,list_u1324408373059187874T_VEBT(Xs,I,Xc),list_u1324408373059187874T_VEBT(Xsi,I,Xi))),F),C2,Q)
           => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_3305_option_Osize_I4_J,axiom,
    ! [X22: nat] : size_size_option_nat(some_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3306_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] : size_s170228958280169651at_nat(some_P7363390416028606310at_nat(X22)) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(4)
tff(fact_3307_option_Osize_I3_J,axiom,
    size_size_option_nat(none_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3308_option_Osize_I3_J,axiom,
    size_s170228958280169651at_nat(none_P5556105721700978146at_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% option.size(3)
tff(fact_3309_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option = some_nat(the_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_3310_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option = some_P7363390416028606310at_nat(the_Pr8591224930841456533at_nat(Option)) ) ) ).

% option.exhaust_sel
tff(fact_3311_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_3312_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_3313_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_3314_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3315_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_3316_div__positive,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ) ) ).

% div_positive
tff(fact_3317_div__positive,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% div_positive
tff(fact_3318_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3319_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(times_times_int(B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_3320_zdiv__mono1,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),A4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),B2)) ) ) ).

% zdiv_mono1
tff(fact_3321_zdiv__mono2,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B4)) ) ) ) ).

% zdiv_mono2
tff(fact_3322_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_3323_zdiv__mono1__neg,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),A4)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_3324_zdiv__mono2__neg,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B4)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_3325_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_3326_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)) ) ) ).

% div_positive_int
tff(fact_3327_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_3328_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_3329_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_3330_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_3331_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_3332_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_3333_div__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_3334_q__pos__lemma,axiom,
    ! [B4: int,Q6: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B4),Q6)),R4))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q6) ) ) ) ).

% q_pos_lemma
tff(fact_3335_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B4: int,Q6: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),Q3)),R3) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B4),Q6)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B4),Q6)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B4)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q6) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3336_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B4: int,Q6: int,R4: int] :
      ( ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),Q3)),R3) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B4),Q6)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B4),Q6)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),B2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3337_unique__quotient__lemma,axiom,
    ! [B2: int,Q6: int,R4: int,Q3: int,R3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),Q6)),R4)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),Q3)),R3))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R4)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q6),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3338_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q6: int,R4: int,Q3: int,R3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),Q6)),R4)),aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),Q3)),R3))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R3),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R3)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q3),Q6) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3339_split__zdiv,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I2: int,J: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J),K)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J) ) )
             => aa_int_o(P,I2) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: int,J: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),zero_zero_int)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J) ) )
             => aa_int_o(P,I2) ) ) ) ) ).

% split_zdiv
tff(fact_3340_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),Q3)),R3) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R3),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R3)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_3341_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),Q3)),R3) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_3342_space__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,Ta)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U)) ) ) ).

% space_bound
tff(fact_3343_space_H__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Ta)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),U)) ) ) ).

% space'_bound
tff(fact_3344_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list_VEBT_VEBT,Xc: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),Xc)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),Nb)) ) ).

% in_children_def
tff(fact_3345_minNull__delete__time__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Ta),Xc))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(Ta),Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ) ).

% minNull_delete_time_bound
tff(fact_3346_tdeletemimi,axiom,
    ! [Deg: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Mia)),Deg,TreeLista),Summarya)),Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% tdeletemimi
tff(fact_3347_vebt__buildup__bound,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),U)) ) ).

% vebt_buildup_bound
tff(fact_3348_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xc: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_3349_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Deg)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xc = Mia )
          | ( Xc = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_3350_T__vebt__buildupi__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% T_vebt_buildupi_univ
tff(fact_3351_T__vebt__buildupi__gq__0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) ).

% T_vebt_buildupi_gq_0
tff(fact_3352_space__space_H,axiom,
    ! [Ta: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_space,Ta)),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Ta)) ).

% space_space'
tff(fact_3353_pure__true,axiom,
    aa_o_assn(pure_assn,$true) = one_one_assn ).

% pure_true
tff(fact_3354_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_assn(pure_assn,(P)) = one_one_assn )
    <=> (P) ) ).

% pure_assn_eq_emp_iff
tff(fact_3355_height__compose__summary,axiom,
    ! [Summarya: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Summarya))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya))) ).

% height_compose_summary
tff(fact_3356_power__one__right,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_3357_power__one__right,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_3358_power__one__right,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_3359_power__one__right,axiom,
    ! [A2: complex] : aa_nat_complex(power_power_complex(A2),one_one_nat) = A2 ).

% power_one_right
tff(fact_3360_power__one__right,axiom,
    ! [A2: code_integer] : power_8256067586552552935nteger(A2,one_one_nat) = A2 ).

% power_one_right
tff(fact_3361_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = one_one_nat )
    <=> ( ( M = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_3362_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) )
    <=> ( ( M = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_3363_ent__pure__pre__iff__sng,axiom,
    ! [B2: $o,Q: assn] :
      ( entails(aa_o_assn(pure_assn,(B2)),Q)
    <=> ( (B2)
       => entails(one_one_assn,Q) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_3364_delete__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_d_e_l_e_t_e(Ta),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))) ) ).

% delete_bound_height
tff(fact_3365_less__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),one_one_nat)
    <=> ( Nb = zero_zero_nat ) ) ).

% less_one
tff(fact_3366_diff__Suc__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),one_one_nat) = Nb ).

% diff_Suc_1
tff(fact_3367_int__div__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = A2 )
      <=> ( B2 = one_one_int ) ) ) ).

% int_div_same_is_1
tff(fact_3368_div__eq__dividend__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = M )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_3369_Suc__diff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_3370_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% Suc_1
tff(fact_3371_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) = Nb ) ) ).

% Suc_diff_1
tff(fact_3372_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),one_one_nat) = Nb ).

% nat_mult_1_right
tff(fact_3373_nat__mult__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),Nb) = Nb ).

% nat_mult_1
tff(fact_3374_assn__one__left,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(one_one_assn),P) = P ).

% assn_one_left
tff(fact_3375_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_3376_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_3377_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),Nb) ).

% Suc_eq_plus1_left
tff(fact_3378_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_3379_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat) ).

% Suc_eq_plus1
tff(fact_3380_nat__geq__1__eq__neqz,axiom,
    ! [Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Xc)
    <=> ( Xc != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_3381_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_3382_mult__eq__self__implies__10,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) )
     => ( ( Nb = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_3383_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(P,N)
               => aa_nat_o(P,aa_nat_nat(suc,N)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_3384_div__less__dividend,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),M) ) ) ).

% div_less_dividend
tff(fact_3385_int__div__less__self,axiom,
    ! [Xc: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xc),K)),Xc) ) ) ).

% int_div_less_self
tff(fact_3386_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% nat_1_add_1
tff(fact_3387_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3388_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_pred'
tff(fact_3389_add__eq__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb) = $ite(M = zero_zero_nat,Nb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_3390_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_3391_mult__eq__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb) = $ite(M = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_3392_power__minus__mult,axiom,
    ! [Nb: nat,A2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A2) = aa_nat_complex(power_power_complex(A2),Nb) ) ) ).

% power_minus_mult
tff(fact_3393_power__minus__mult,axiom,
    ! [Nb: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( times_3573771949741848930nteger(power_8256067586552552935nteger(A2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),A2) = power_8256067586552552935nteger(A2,Nb) ) ) ).

% power_minus_mult
tff(fact_3394_power__minus__mult,axiom,
    ! [Nb: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A2) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb) ) ) ).

% power_minus_mult
tff(fact_3395_power__minus__mult,axiom,
    ! [Nb: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A2) = aa_nat_rat(power_power_rat(A2),Nb) ) ) ).

% power_minus_mult
tff(fact_3396_power__minus__mult,axiom,
    ! [Nb: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb) ) ) ).

% power_minus_mult
tff(fact_3397_power__minus__mult,axiom,
    ! [Nb: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(times_times_int(aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A2) = aa_nat_int(power_power_int(A2),Nb) ) ) ).

% power_minus_mult
tff(fact_3398_power__minus__mult,axiom,
    ! [Nb: nat,A2: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_assn_assn(times_times_assn(aa_nat_assn(power_power_assn(A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A2) = aa_nat_assn(power_power_assn(A2),Nb) ) ) ).

% power_minus_mult
tff(fact_3399_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_3400_axxdiv2,axiom,
    ! [Xc: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Xc)),Xc)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = Xc )
      & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(zero_zero_int),Xc)),Xc)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = Xc ) ) ).

% axxdiv2
tff(fact_3401_z1pdiv2,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = B2 ).

% z1pdiv2
tff(fact_3402_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_3403_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_3404_small__powers__of__2,axiom,
    ! [Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Xc)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xc),one_one_nat))) ) ).

% small_powers_of_2
tff(fact_3405_power__2__mult__step__le,axiom,
    ! [N4: nat,Nb: nat,K4: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N4),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)),K4)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K4),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_3406_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_3407_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(B2),one_one_int)),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_3408_tdeletemimi_H,axiom,
    ! [Deg: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Mia)),Deg,TreeLista),Summarya)),Xc)),one_one_nat) ) ).

% tdeletemimi'
tff(fact_3409_Tb__T__vebt__buildupi_H_H,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(vEBT_VEBT_Tb2,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi''
tff(fact_3410_minNull__delete__time__bound_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(Ta),Xc))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Ta),Xc)),one_one_nat) ) ) ).

% minNull_delete_time_bound'
tff(fact_3411_delete__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Ta),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).

% delete_bound_height'
tff(fact_3412_divmod__step__eq,axiom,
    ! [L: num,Q3: nat,R3: nat] :
      unique5026877609467782581ep_nat(L,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R3)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,L)),R3),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q3)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R3),aa_num_nat(numeral_numeral_nat,L))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Q3)),R3)) ).

% divmod_step_eq
tff(fact_3413_divmod__step__eq,axiom,
    ! [L: num,Q3: int,R3: int] :
      unique5024387138958732305ep_int(L,product_Pair_int_int(Q3,R3)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,L)),R3),product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3)),one_one_int),aa_int_int(minus_minus_int(R3),aa_num_int(numeral_numeral_int,L))),product_Pair_int_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q3),R3)) ).

% divmod_step_eq
tff(fact_3414_divmod__step__eq,axiom,
    ! [L: num,Q3: code_integer,R3: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q3,R3)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(L)),R3),produc1086072967326762835nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Q3),one_one_Code_integer),minus_8373710615458151222nteger(R3,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Q3),R3)) ).

% divmod_step_eq
tff(fact_3415_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: $o,F2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q)
    <=> ( (B2)
       => hoare_hoare_triple_o(P,F2,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_3416_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: $o,F2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q)
    <=> ( (B2)
       => hoare_1429296392585015714_VEBTi(P,F2,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_3417_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: $o,F2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q)
    <=> ( (B2)
       => hoare_7629718768684598413on_nat(P,F2,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_3418_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: $o,F2: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q)
    <=> ( (B2)
       => hoare_3067605981109127869le_nat(P,F2,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_3419_space__2__pow__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Ta))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)),one_one_real))) ) ).

% space_2_pow_bound
tff(fact_3420_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(plus_plus_int(Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_3421_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_3422_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_3423_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_3424_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_3425_of__nat__eq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( M = Nb ) ) ).

% of_nat_eq_iff
tff(fact_3426_two__realpow__ge__two,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ) ).

% two_realpow_ge_two
tff(fact_3427_double__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% double_eq_0_iff
tff(fact_3428_double__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_3429_double__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( aa_int_int(plus_plus_int(A2),A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_3430_count__buildup_H,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) ).

% count_buildup'
tff(fact_3431_space__cnt,axiom,
    ! [Ta: vEBT_VEBT] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Ta))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Ta))) ).

% space_cnt
tff(fact_3432_T__vebt__buildupi__cnt_H,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V441764108873111860ildupi,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb)))) ).

% T_vebt_buildupi_cnt'
tff(fact_3433_t__buildup__cnt,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))) ).

% t_buildup_cnt
tff(fact_3434_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% semiring_1_class.of_nat_0
tff(fact_3435_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% semiring_1_class.of_nat_0
tff(fact_3436_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% semiring_1_class.of_nat_0
tff(fact_3437_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% semiring_1_class.of_nat_0
tff(fact_3438_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% semiring_1_class.of_nat_0
tff(fact_3439_semiring__1__class_Oof__nat__0,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,zero_zero_nat) = zero_z3403309356797280102nteger ).

% semiring_1_class.of_nat_0
tff(fact_3440_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_3441_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_3442_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_3443_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_3444_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_zero_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_3445_of__nat__0__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( zero_z3403309356797280102nteger = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( zero_zero_nat = Nb ) ) ).

% of_nat_0_eq_iff
tff(fact_3446_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,M) = zero_zero_rat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3447_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = zero_zero_real )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3448_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = zero_zero_int )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3449_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,M) = zero_zero_nat )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3450_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,M) = zero_zero_complex )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3451_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,M) = zero_z3403309356797280102nteger )
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_3452_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_rat(semiri681578069525770553at_rat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% of_nat_numeral
tff(fact_3453_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_real(semiri5074537144036343181t_real,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% of_nat_numeral
tff(fact_3454_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% of_nat_numeral
tff(fact_3455_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_nat(semiri1316708129612266289at_nat,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,Nb) ).

% of_nat_numeral
tff(fact_3456_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Nb)) = numera6690914467698888265omplex(Nb) ).

% of_nat_numeral
tff(fact_3457_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_num_nat(numeral_numeral_nat,Nb)) = numera6620942414471956472nteger(Nb) ).

% of_nat_numeral
tff(fact_3458_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_3459_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_3460_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_3461_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_3462_of__nat__less__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_iff
tff(fact_3463_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_3464_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_3465_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_3466_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_3467_of__nat__le__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% of_nat_le_iff
tff(fact_3468_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_3469_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_3470_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_3471_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_3472_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% of_nat_add
tff(fact_3473_of__nat__add,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) = plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_add
tff(fact_3474_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_3475_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_3476_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_3477_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_3478_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_3479_of__nat__1,axiom,
    aa_nat_Code_integer(semiri4939895301339042750nteger,one_one_nat) = one_one_Code_integer ).

% of_nat_1
tff(fact_3480_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_rat = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3481_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_real = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3482_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_int = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3483_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3484_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_complex = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3485_of__nat__1__eq__iff,axiom,
    ! [Nb: nat] :
      ( ( one_one_Code_integer = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_1_eq_iff
tff(fact_3486_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Nb) = one_one_rat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3487_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Nb) = one_one_real )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3488_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Nb) = one_one_int )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3489_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Nb) = one_one_nat )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3490_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Nb) = one_one_complex )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3491_of__nat__eq__1__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) = one_one_Code_integer )
    <=> ( Nb = one_one_nat ) ) ).

% of_nat_eq_1_iff
tff(fact_3492_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_mult
tff(fact_3493_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_3494_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_3495_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_3496_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% of_nat_mult
tff(fact_3497_of__nat__mult,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)) = times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mult
tff(fact_3498_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_3499_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_3500_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_3501_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_3502_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,M),Nb)) = power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),Nb) ).

% semiring_1_class.of_nat_power
tff(fact_3503_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W) = aa_nat_real(semiri5074537144036343181t_real,Xc) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xc ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3504_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W) = aa_nat_int(semiri1314217659103216013at_int,Xc) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xc ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3505_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W) = aa_nat_nat(semiri1316708129612266289at_nat,Xc) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xc ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3506_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B2)),W) = aa_nat_complex(semiri8010041392384452111omplex,Xc) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xc ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3507_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( ( power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2),W) = aa_nat_Code_integer(semiri4939895301339042750nteger,Xc) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xc ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_3508_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Xc) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W) )
    <=> ( Xc = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3509_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Xc) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W) )
    <=> ( Xc = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3510_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Xc) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W) )
    <=> ( Xc = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3511_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Xc) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B2)),W) )
    <=> ( Xc = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3512_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Xc) = power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2),W) )
    <=> ( Xc = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_3513_norm__pre__pure__iff__sng,axiom,
    ! [B2: $o,F2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( hoare_hoare_triple_o(aa_o_assn(pure_assn,(B2)),F2,Q)
    <=> ( (B2)
       => hoare_hoare_triple_o(one_one_assn,F2,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_3514_norm__pre__pure__iff__sng,axiom,
    ! [B2: $o,F2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(aa_o_assn(pure_assn,(B2)),F2,Q)
    <=> ( (B2)
       => hoare_1429296392585015714_VEBTi(one_one_assn,F2,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_3515_norm__pre__pure__iff__sng,axiom,
    ! [B2: $o,F2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(aa_o_assn(pure_assn,(B2)),F2,Q)
    <=> ( (B2)
       => hoare_7629718768684598413on_nat(one_one_assn,F2,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_3516_norm__pre__pure__iff__sng,axiom,
    ! [B2: $o,F2: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(aa_o_assn(pure_assn,(B2)),F2,Q)
    <=> ( (B2)
       => hoare_3067605981109127869le_nat(one_one_assn,F2,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_3517_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3518_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3519_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3520_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3521_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int)
    <=> ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_3522_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,M)) = aa_rat_rat(plus_plus_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M)) ).

% of_nat_Suc
tff(fact_3523_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M)) ).

% of_nat_Suc
tff(fact_3524_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,M)) = aa_int_int(plus_plus_int(one_one_int),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% of_nat_Suc
tff(fact_3525_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,M)) ).

% of_nat_Suc
tff(fact_3526_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,M)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,M)) ).

% of_nat_Suc
tff(fact_3527_of__nat__Suc,axiom,
    ! [M: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,M)) = plus_p5714425477246183910nteger(one_one_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)) ).

% of_nat_Suc
tff(fact_3528_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),aa_int_int(minus_minus_int(Z),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% zle_diff1_eq
tff(fact_3529_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_3530_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_3531_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_3532_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_3533_of__nat__0__less__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% of_nat_0_less_iff
tff(fact_3534_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xc)),Nb) = aa_nat_rat(semiri681578069525770553at_rat,Ya) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3535_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xc)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Ya) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3536_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) = aa_nat_int(semiri1314217659103216013at_int,Ya) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3537_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) = aa_nat_nat(semiri1316708129612266289at_nat,Ya) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3538_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xc)),Nb) = aa_nat_complex(semiri8010041392384452111omplex,Ya) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3539_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: nat] :
      ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(Xc),Nb) = aa_nat_Code_integer(semiri4939895301339042750nteger,Ya) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_3540_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: nat,Xc: num,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Ya) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xc)),Nb) )
    <=> ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3541_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: nat,Xc: num,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Ya) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xc)),Nb) )
    <=> ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3542_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: nat,Xc: num,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Ya) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) )
    <=> ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3543_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: nat,Xc: num,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Ya) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) )
    <=> ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3544_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: nat,Xc: num,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Ya) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xc)),Nb) )
    <=> ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3545_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: nat,Xc: num,Nb: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,Ya) = power_8256067586552552935nteger(numera6620942414471956472nteger(Xc),Nb) )
    <=> ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_3546_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xc) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3547_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W)),aa_nat_real(semiri5074537144036343181t_real,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xc) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3548_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W)),aa_nat_int(semiri1314217659103216013at_int,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xc) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3549_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xc) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3550_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xc) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_3551_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xc)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3552_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3553_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xc)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3554_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3555_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xc)),power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_3556_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W)),aa_nat_real(semiri5074537144036343181t_real,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xc) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3557_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2),W)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xc) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3558_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xc) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3559_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xc) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3560_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xc: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W)),aa_nat_int(semiri1314217659103216013at_int,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xc) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_3561_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3562_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xc)),power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,B2),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3563_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xc)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3564_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3565_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xc: nat,B2: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xc)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_3566_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,W)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_3567_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_num_real(numeral_numeral_real,W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(numeral_numeral_nat,W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_3568_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,M))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_3569_of__nat__zero__less__power__iff,axiom,
    ! [Xc: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xc)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xc)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3570_of__nat__zero__less__power__iff,axiom,
    ! [Xc: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xc)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xc)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3571_of__nat__zero__less__power__iff,axiom,
    ! [Xc: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xc)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xc)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3572_of__nat__zero__less__power__iff,axiom,
    ! [Xc: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xc)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xc)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3573_of__nat__zero__less__power__iff,axiom,
    ! [Xc: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xc),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xc)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_3574_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xc: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xc) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3575_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xc: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xc) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3576_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xc: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xc) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3577_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xc) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3578_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xc: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(numera6620942414471956472nteger(I),Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xc) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_3579_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xc: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xc)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3580_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xc: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3581_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xc: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xc)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3582_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xc: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3583_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xc: nat,I: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xc)),power_8256067586552552935nteger(numera6620942414471956472nteger(I),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_3584_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xc: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xc) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3585_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xc: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(numera6620942414471956472nteger(I),Nb)),aa_nat_Code_integer(semiri4939895301339042750nteger,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xc) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3586_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xc: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xc) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3587_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xc) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3588_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xc: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xc))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xc) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_3589_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xc: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3590_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xc: nat,I: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Xc)),power_8256067586552552935nteger(numera6620942414471956472nteger(I),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3591_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xc: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xc)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3592_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xc: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3593_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xc: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xc)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_3594_mult__of__nat__commute,axiom,
    ! [Xc: nat,Ya: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,Xc)),Ya) = aa_rat_rat(times_times_rat(Ya),aa_nat_rat(semiri681578069525770553at_rat,Xc)) ).

% mult_of_nat_commute
tff(fact_3595_mult__of__nat__commute,axiom,
    ! [Xc: nat,Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Xc)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),aa_nat_real(semiri5074537144036343181t_real,Xc)) ).

% mult_of_nat_commute
tff(fact_3596_mult__of__nat__commute,axiom,
    ! [Xc: nat,Ya: int] : aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,Xc)),Ya) = aa_int_int(times_times_int(Ya),aa_nat_int(semiri1314217659103216013at_int,Xc)) ).

% mult_of_nat_commute
tff(fact_3597_mult__of__nat__commute,axiom,
    ! [Xc: nat,Ya: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xc)),Ya) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Ya),aa_nat_nat(semiri1316708129612266289at_nat,Xc)) ).

% mult_of_nat_commute
tff(fact_3598_mult__of__nat__commute,axiom,
    ! [Xc: nat,Ya: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Xc)),Ya) = aa_complex_complex(times_times_complex(Ya),aa_nat_complex(semiri8010041392384452111omplex,Xc)) ).

% mult_of_nat_commute
tff(fact_3599_mult__of__nat__commute,axiom,
    ! [Xc: nat,Ya: code_integer] : times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,Xc),Ya) = times_3573771949741848930nteger(Ya,aa_nat_Code_integer(semiri4939895301339042750nteger,Xc)) ).

% mult_of_nat_commute
tff(fact_3600_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_0_le_iff
tff(fact_3601_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_0_le_iff
tff(fact_3602_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_0_le_iff
tff(fact_3603_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_0_le_iff
tff(fact_3604_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_0_le_iff
tff(fact_3605_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_3606_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_3607_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_3608_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_3609_of__nat__less__0__iff,axiom,
    ! [M: nat] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),zero_z3403309356797280102nteger) ).

% of_nat_less_0_iff
tff(fact_3610_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3611_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3612_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3613_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3614_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb)) != zero_zero_complex ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3615_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(suc,Nb)) != zero_z3403309356797280102nteger ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_3616_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_3617_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_3618_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_3619_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_3620_less__imp__of__nat__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_3621_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_3622_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_3623_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_3624_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_3625_of__nat__less__imp__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ) ).

% of_nat_less_imp_less
tff(fact_3626_div__mult2__eq_H,axiom,
    ! [A2: int,M: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% div_mult2_eq'
tff(fact_3627_div__mult2__eq_H,axiom,
    ! [A2: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% div_mult2_eq'
tff(fact_3628_div__mult2__eq_H,axiom,
    ! [A2: code_integer,M: nat,Nb: nat] : divide6298287555418463151nteger(A2,times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = divide6298287555418463151nteger(divide6298287555418463151nteger(A2,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% div_mult2_eq'
tff(fact_3629_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J2)) ) ).

% of_nat_mono
tff(fact_3630_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,I)),aa_nat_Code_integer(semiri4939895301339042750nteger,J2)) ) ).

% of_nat_mono
tff(fact_3631_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J2)) ) ).

% of_nat_mono
tff(fact_3632_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J2)) ) ).

% of_nat_mono
tff(fact_3633_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J2)) ) ).

% of_nat_mono
tff(fact_3634_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3635_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3636_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = divide6298287555418463151nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_3637_of__nat__max,axiom,
    ! [Xc: nat,Ya: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_nat_real(semiri5074537144036343181t_real,Xc)),aa_nat_real(semiri5074537144036343181t_real,Ya)) ).

% of_nat_max
tff(fact_3638_of__nat__max,axiom,
    ! [Xc: nat,Ya: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),Ya)) = aa_int_int(ord_max_int(aa_nat_int(semiri1314217659103216013at_int,Xc)),aa_nat_int(semiri1314217659103216013at_int,Ya)) ).

% of_nat_max
tff(fact_3639_of__nat__max,axiom,
    ! [Xc: nat,Ya: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xc)),aa_nat_nat(semiri1316708129612266289at_nat,Ya)) ).

% of_nat_max
tff(fact_3640_of__nat__max,axiom,
    ! [Xc: nat,Ya: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),Ya)) = ord_max_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Xc),aa_nat_Code_integer(semiri4939895301339042750nteger,Ya)) ).

% of_nat_max
tff(fact_3641_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ) ).

% of_nat_diff
tff(fact_3642_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_3643_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_3644_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_3645_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ) ) ).

% of_nat_diff
tff(fact_3646_of__nat__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) = minus_8373710615458151222nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ) ) ).

% of_nat_diff
tff(fact_3647_reals__Archimedean3,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ! [Y3: real] :
        ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xc)) ) ).

% reals_Archimedean3
tff(fact_3648_real__of__nat__div4,axiom,
    ! [Nb: nat,Xc: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xc))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xc))) ).

% real_of_nat_div4
tff(fact_3649_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(minus_minus_int(K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_3650_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I5: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I5),K)
             => ( aa_int_o(P,I5)
               => aa_int_o(P,aa_int_int(minus_minus_int(I5),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_3651_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,aa_int_int(minus_minus_int(K),one_one_int))
       => ( ! [I5: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I5),K)
             => ( aa_int_o(P,I5)
               => aa_int_o(P,aa_int_int(minus_minus_int(I5),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_3652_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(times_times_int(aa_int_int(minus_minus_int(Z1),Z22)),W) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(Z1),W)),aa_int_int(times_times_int(Z22),W)) ).

% int_distrib(3)
tff(fact_3653_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(times_times_int(W),aa_int_int(minus_minus_int(Z1),Z22)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(W),Z1)),aa_int_int(times_times_int(W),Z22)) ).

% int_distrib(4)
tff(fact_3654_nat__less__real__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,M)) ) ).

% nat_less_real_le
tff(fact_3655_nat__le__real__less,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,M)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_3656_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_3657_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_3658_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_3659_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) ).

% of_nat_less_two_power
tff(fact_3660_inverse__of__nat__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( ( Nb != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,M))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_3661_inverse__of__nat__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( ( Nb != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,M))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_3662_real__archimedian__rdiv__eq__0,axiom,
    ! [Xc: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C2)
       => ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M4)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M4)),Xc)),C2) )
         => ( Xc = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_3663_real__of__nat__div2,axiom,
    ! [Nb: nat,Xc: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xc))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xc)))) ).

% real_of_nat_div2
tff(fact_3664_real__of__nat__div3,axiom,
    ! [Nb: nat,Xc: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xc))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xc)))),one_one_real) ).

% real_of_nat_div3
tff(fact_3665_less__int__code_I1_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),zero_zero_int) ).

% less_int_code(1)
tff(fact_3666_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I5: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I5)
           => ( aa_int_o(P,I5)
             => aa_int_o(P,aa_int_int(plus_plus_int(I5),one_one_int)) ) )
       => ( ! [I5: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I5),K)
             => ( aa_int_o(P,I5)
               => aa_int_o(P,aa_int_int(minus_minus_int(I5),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_3667_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,aa_int_int(plus_plus_int(K),one_one_int))
       => ( ! [I5: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I5)
             => ( aa_int_o(P,I5)
               => aa_int_o(P,aa_int_int(plus_plus_int(I5),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_3668_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(plus_plus_int(Z),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_3669_cons__post__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_o,Q: fun_o_assn,Q2: fun_o_assn] :
      ( hoare_hoare_triple_o(P,C2,Q)
     => ( ! [X3: $o] : entails(aa_o_assn(Q,(X3)),aa_o_assn(Q2,(X3)))
       => hoare_hoare_triple_o(P,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_3670_cons__post__rule,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Q2: fun_VEBT_VEBTi_assn] :
      ( hoare_1429296392585015714_VEBTi(P,C2,Q)
     => ( ! [X3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q,X3),aa_VEBT_VEBTi_assn(Q2,X3))
       => hoare_1429296392585015714_VEBTi(P,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_3671_cons__post__rule,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Q2: fun_option_nat_assn] :
      ( hoare_7629718768684598413on_nat(P,C2,Q)
     => ( ! [X3: option_nat] : entails(aa_option_nat_assn(Q,X3),aa_option_nat_assn(Q2,X3))
       => hoare_7629718768684598413on_nat(P,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_3672_cons__post__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,Q2: fun_nat_assn] :
      ( hoare_3067605981109127869le_nat(P,C2,Q)
     => ( ! [X3: nat] : entails(aa_nat_assn(Q,X3),aa_nat_assn(Q2,X3))
       => hoare_3067605981109127869le_nat(P,C2,Q2) ) ) ).

% cons_post_rule
tff(fact_3673_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_o_assn,Q2: fun_o_assn,C2: heap_Time_Heap_o] :
      ( entails(P,P2)
     => ( ! [X3: $o] : entails(aa_o_assn(Q,(X3)),aa_o_assn(Q2,(X3)))
       => ( hoare_hoare_triple_o(P2,C2,Q)
         => hoare_hoare_triple_o(P,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_3674_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_VEBT_VEBTi_assn,Q2: fun_VEBT_VEBTi_assn,C2: heap_T8145700208782473153_VEBTi] :
      ( entails(P,P2)
     => ( ! [X3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q,X3),aa_VEBT_VEBTi_assn(Q2,X3))
       => ( hoare_1429296392585015714_VEBTi(P2,C2,Q)
         => hoare_1429296392585015714_VEBTi(P,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_3675_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_option_nat_assn,Q2: fun_option_nat_assn,C2: heap_T2636463487746394924on_nat] :
      ( entails(P,P2)
     => ( ! [X3: option_nat] : entails(aa_option_nat_assn(Q,X3),aa_option_nat_assn(Q2,X3))
       => ( hoare_7629718768684598413on_nat(P2,C2,Q)
         => hoare_7629718768684598413on_nat(P,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_3676_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: fun_nat_assn,Q2: fun_nat_assn,C2: heap_Time_Heap_nat] :
      ( entails(P,P2)
     => ( ! [X3: nat] : entails(aa_nat_assn(Q,X3),aa_nat_assn(Q2,X3))
       => ( hoare_3067605981109127869le_nat(P2,C2,Q)
         => hoare_3067605981109127869le_nat(P,C2,Q2) ) ) ) ).

% cons_rule
tff(fact_3677_norm__pre__pure__rule2,axiom,
    ! [B2: $o,F2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( ( (B2)
       => hoare_hoare_triple_o(one_one_assn,F2,Q) )
     => hoare_hoare_triple_o(aa_o_assn(pure_assn,(B2)),F2,Q) ) ).

% norm_pre_pure_rule2
tff(fact_3678_norm__pre__pure__rule2,axiom,
    ! [B2: $o,F2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ( (B2)
       => hoare_1429296392585015714_VEBTi(one_one_assn,F2,Q) )
     => hoare_1429296392585015714_VEBTi(aa_o_assn(pure_assn,(B2)),F2,Q) ) ).

% norm_pre_pure_rule2
tff(fact_3679_norm__pre__pure__rule2,axiom,
    ! [B2: $o,F2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ( (B2)
       => hoare_7629718768684598413on_nat(one_one_assn,F2,Q) )
     => hoare_7629718768684598413on_nat(aa_o_assn(pure_assn,(B2)),F2,Q) ) ).

% norm_pre_pure_rule2
tff(fact_3680_norm__pre__pure__rule2,axiom,
    ! [B2: $o,F2: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ( (B2)
       => hoare_3067605981109127869le_nat(one_one_assn,F2,Q) )
     => hoare_3067605981109127869le_nat(aa_o_assn(pure_assn,(B2)),F2,Q) ) ).

% norm_pre_pure_rule2
tff(fact_3681_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_3682_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Z)),Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_3683_zmult__zless__mono2,axiom,
    ! [I: int,J2: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(K),I)),aa_int_int(times_times_int(K),J2)) ) ) ).

% zmult_zless_mono2
tff(fact_3684_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( ( aa_int_int(times_times_int(M),Nb) = one_one_int )
      <=> ( ( M = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3685_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(W),one_one_int)),Z)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% add1_zle_eq
tff(fact_3686_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(W),one_one_int)),Z) ) ).

% zless_imp_add1_zle
tff(fact_3687_norm__pre__pure__rule1,axiom,
    ! [B2: $o,P: assn,F2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( ( (B2)
       => hoare_hoare_triple_o(P,F2,Q) )
     => hoare_hoare_triple_o(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q) ) ).

% norm_pre_pure_rule1
tff(fact_3688_norm__pre__pure__rule1,axiom,
    ! [B2: $o,P: assn,F2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( ( (B2)
       => hoare_1429296392585015714_VEBTi(P,F2,Q) )
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q) ) ).

% norm_pre_pure_rule1
tff(fact_3689_norm__pre__pure__rule1,axiom,
    ! [B2: $o,P: assn,F2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( ( (B2)
       => hoare_7629718768684598413on_nat(P,F2,Q) )
     => hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q) ) ).

% norm_pre_pure_rule1
tff(fact_3690_norm__pre__pure__rule1,axiom,
    ! [B2: $o,P: assn,F2: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( ( (B2)
       => hoare_3067605981109127869le_nat(P,F2,Q) )
     => hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q) ) ).

% norm_pre_pure_rule1
tff(fact_3691_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),Z)) ) ).

% le_imp_0_less
tff(fact_3692_Tb_H__cnt,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_VEBT_Tb2,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb)))) ).

% Tb'_cnt
tff(fact_3693_cnt__cnt__eq,axiom,
    ! [Ta: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Ta) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Ta)) ).

% cnt_cnt_eq
tff(fact_3694_t__build__cnt,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))) ).

% t_build_cnt
tff(fact_3695_height__node,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya))) ) ).

% height_node
tff(fact_3696_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),TreeLista),Summarya)),Xc) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_3697_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),TreeLista),Summarya)),Xc) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_3698_linear__plus__1__le__power,axiom,
    ! [Xc: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xc)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_3699_delete__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_d_e_l_e_t_e(Ta),Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% delete_bound_size_univ
tff(fact_3700_buildup__build__time,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,Nb)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Nb)) ).

% buildup_build_time
tff(fact_3701_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,M) = aa_num_int(numeral_numeral_int,V) )
    <=> ( M = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_3702_delete__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Ta),Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).

% delete_bound_size_univ'
tff(fact_3703_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,Ta: vEBT_VEBT] :
      ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Deg) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Deg)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_3704_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N: nat] : Z != aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,M4)),aa_nat_int(semiri1314217659103216013at_int,N)) ).

% int_diff_cases
tff(fact_3705_zle__int,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% zle_int
tff(fact_3706_zadd__int__left,axiom,
    ! [M: nat,Nb: nat,Z: int] : aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),Z)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))),Z) ).

% zadd_int_left
tff(fact_3707_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% zdiv_int
tff(fact_3708_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z)
    <=> ? [N6: nat] : Z = aa_int_int(plus_plus_int(W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N6))) ) ).

% zless_iff_Suc_zadd
tff(fact_3709_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_3710_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% pos_int_cases
tff(fact_3711_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J2: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,K)),J2)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_3712_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( ! [A3: real,B3: real,C4: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A3),B3)
           => ( aa_real_o(aa_real_fun_real_o(P,B3),C4)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),B3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),C4)
                 => aa_real_o(aa_real_fun_real_o(P,A3),C4) ) ) ) )
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2)
               => ? [D4: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D4)
                    & ! [A3: real,B3: real] :
                        ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A3),X3)
                          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B3)
                          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B3),A3)),D4) )
                       => aa_real_o(aa_real_fun_real_o(P,A3),B3) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A2),B2) ) ) ) ).

% Bolzano
tff(fact_3713_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Uu: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya)),Uu) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_3714_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Uu: nat] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya)),Uu) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
tff(fact_3715_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,TreeLista),Summarya)),Xc) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_3716_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,TreeLista),Summarya)),Xc) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_3717_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),B2)) = aa_nat_real(semiri5074537144036343181t_real,B2) ) ) ) ).

% log_pow_cancel
tff(fact_3718_Tb__T__vebt__buildupi,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,Nb))),aa_int_int(minus_minus_int(aa_nat_int(vEBT_VEBT_Tb,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi
tff(fact_3719_zero__le__log__cancel__iff,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_3720_log__le__zero__cancel__iff,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_3721_one__le__log__cancel__iff,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Xc) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_3722_log__le__one__cancel__iff,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),A2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_3723_log__le__cancel__iff,axiom,
    ! [A2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Ya))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_3724_log2__of__power__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_3725_Tb__Tb_H,axiom,
    ! [Ta: nat] : aa_nat_int(vEBT_VEBT_Tb,Ta) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_VEBT_Tb2,Ta)) ).

% Tb_Tb'
tff(fact_3726_log__one,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A2),one_one_real) = zero_zero_real ).

% log_one
tff(fact_3727_log__eq__one,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),A2) = one_one_real ) ) ) ).

% log_eq_one
tff(fact_3728_log__less__cancel__iff,axiom,
    ! [A2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Ya))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_3729_log__less__one__cancel__iff,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),A2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_3730_one__less__log__cancel__iff,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Xc) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_3731_log__less__zero__cancel__iff,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_3732_zero__less__log__cancel__iff,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_3733_log__base__change,axiom,
    ! [A2: real,B2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),B2)) ) ) ) ).

% log_base_change
tff(fact_3734_less__log__of__power,axiom,
    ! [B2: real,Nb: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),M)) ) ) ).

% less_log_of_power
tff(fact_3735_log__of__power__eq,axiom,
    ! [M: nat,B2: real,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,M) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ) ).

% log_of_power_eq
tff(fact_3736_log__mult,axiom,
    ! [A2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Ya)) ) ) ) ) ) ).

% log_mult
tff(fact_3737_le__log__of__power,axiom,
    ! [B2: real,Nb: nat,M: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb)),M)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),M)) ) ) ).

% le_log_of_power
tff(fact_3738_log__divide,axiom,
    ! [A2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Ya)) ) ) ) ) ) ).

% log_divide
tff(fact_3739_log__base__pow,axiom,
    ! [A2: real,Nb: nat,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),Xc) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log_base_pow
tff(fact_3740_log__nat__power,axiom,
    ! [Xc: real,B2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)) ) ) ).

% log_nat_power
tff(fact_3741_log2__of__power__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M)) ) ) ).

% log2_of_power_eq
tff(fact_3742_log__of__power__less,axiom,
    ! [M: nat,B2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_3743_log__of__power__le,axiom,
    ! [M: nat,B2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_3744_less__log2__of__power,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% less_log2_of_power
tff(fact_3745_le__log2__of__power,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),M)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ).

% le_log2_of_power
tff(fact_3746_log2__of__power__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_3747_Tb__T__vebt__buildupi_H,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(vEBT_V9176841429113362141ildupi,Nb)),aa_int_int(minus_minus_int(aa_nat_int(vEBT_VEBT_Tb,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% Tb_T_vebt_buildupi'
tff(fact_3748_Tbuildupi__buildupi_H,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) = aa_nat_int(vEBT_V9176841429113362141ildupi,Nb) ).

% Tbuildupi_buildupi'
tff(fact_3749_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_3750_setprop,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Ta),set_VEBT_VEBT2(treeList))
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,na),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% setprop
tff(fact_3751_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista),Summarya)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xc),Xc)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista),Summarya) ).

% vebt_insert.simps(4)
tff(fact_3752_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi),Vj) = none_nat ).

% vebt_pred.simps(6)
tff(fact_3753_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh),Vi) = none_nat ).

% vebt_succ.simps(5)
tff(fact_3754_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_3755_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),one_one_nat)
    <=> ? [A6: $o,B6: $o] : Ta = aa_o_VEBT_VEBT(vEBT_Leaf((A6)),(B6)) ) ).

% deg1Leaf
tff(fact_3756_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),one_one_nat)
     => ? [A3: $o,B3: $o] : Ta = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) ) ).

% deg_1_Leaf
tff(fact_3757_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( Nb = one_one_nat )
       => ? [A3: $o,B3: $o] : Ta = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_3758_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Nb: nat] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),set_int2(Xs))
         => aa_int_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
       => aa_int_o(P,aa_nat_int(nth_int(Xs),Nb)) ) ) ).

% inthall
tff(fact_3759_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Nb: nat] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),set_complex2(Xs))
         => aa_complex_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
       => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),Nb)) ) ) ).

% inthall
tff(fact_3760_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Nb: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
       => aa_VEBT_VEBT_o(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)) ) ) ).

% inthall
tff(fact_3761_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Nb: nat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_real2(Xs))
         => aa_real_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% inthall
tff(fact_3762_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Nb: nat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),set_o2(Xs))
         => aa_o_o(P,(X3)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% inthall
tff(fact_3763_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_nat_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% inthall
tff(fact_3764_inthall,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o,Nb: nat] :
      ( ! [X3: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(Xs))
         => aa_VEBT_VEBTi_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
       => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% inthall
tff(fact_3765_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
      ( ( aa_o_VEBT_VEBT(vEBT_Leaf((X21)),(X222)) = aa_o_VEBT_VEBT(vEBT_Leaf((Y21)),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X222)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_3766_height__compose__child,axiom,
    ! [Ta: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summarya: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Ta),set_VEBT_VEBT2(TreeLista))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya))) ) ).

% height_compose_child
tff(fact_3767_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Deg)
     => ( ( Mia = Maa )
       => ( ! [X4: vEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_13) )
          & ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),X_13) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_3768_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),Nb) )
     => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_3769_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_s6755466524823107622T_VEBT(Xs))
       => ( set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(list_u1324408373059187874T_VEBT(Xs,I,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),J2)),J2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I))) = set_VEBT_VEBT2(Xs) ) ) ) ).

% set_swap
tff(fact_3770_set__swap,axiom,
    ! [I: nat,Xs: list_real,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_real(Xs))
       => ( set_real2(list_update_real(list_update_real(Xs,I,aa_nat_real(nth_real(Xs),J2)),J2,aa_nat_real(nth_real(Xs),I))) = set_real2(Xs) ) ) ) ).

% set_swap
tff(fact_3771_set__swap,axiom,
    ! [I: nat,Xs: list_o,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_o(Xs))
       => ( set_o2(list_update_o(list_update_o(Xs,I,aa_nat_o(nth_o(Xs),J2)),J2,aa_nat_o(nth_o(Xs),I))) = set_o2(Xs) ) ) ) ).

% set_swap
tff(fact_3772_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_size_list_nat(Xs))
       => ( set_nat2(list_update_nat(list_update_nat(Xs,I,aa_nat_nat(nth_nat(Xs),J2)),J2,aa_nat_nat(nth_nat(Xs),I))) = set_nat2(Xs) ) ) ) ).

% set_swap
tff(fact_3773_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),size_s7982070591426661849_VEBTi(Xs))
       => ( set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(list_u6098035379799741383_VEBTi(Xs,I,nth_VEBT_VEBTi(Xs,J2)),J2,nth_VEBT_VEBTi(Xs,I))) = set_VEBT_VEBTi2(Xs) ) ) ) ).

% set_swap
tff(fact_3774_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,aa_o_VEBT_VEBT(vEBT_Leaf((X21)),(X222))) = zero_zero_nat ).

% VEBT.size(4)
tff(fact_3775_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xc: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv: $o,D5: nat] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)),D5)
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mima,Deg2,TreeList2),Summary),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_3776_VEBT_Oexhaust,axiom,
    ! [Ya: vEBT_VEBT] :
      ( ! [X11: option4927543243414619207at_nat,X122: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : Ya != aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11,X122,X13),X14)
     => ~ ! [X212: $o,X223: $o] : Ya != aa_o_VEBT_VEBT(vEBT_Leaf((X212)),(X223)) ) ).

% VEBT.exhaust
tff(fact_3777_VEBT_Odistinct_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT,X21: $o,X222: $o] : aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12,X13a),X14a) != aa_o_VEBT_VEBT(vEBT_Leaf((X21)),(X222)) ).

% VEBT.distinct(1)
tff(fact_3778_subset__code_I1_J,axiom,
    ! [Xs: list_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),B)
    <=> ! [X2: int] :
          ( aa_set_int_o(member_int(X2),set_int2(Xs))
         => aa_set_int_o(member_int(X2),B) ) ) ).

% subset_code(1)
tff(fact_3779_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),B)
    <=> ! [X2: complex] :
          ( aa_set_complex_o(member_complex(X2),set_complex2(Xs))
         => aa_set_complex_o(member_complex(X2),B) ) ) ).

% subset_code(1)
tff(fact_3780_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),B)
    <=> ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),B) ) ) ).

% subset_code(1)
tff(fact_3781_subset__code_I1_J,axiom,
    ! [Xs: list_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),B)
    <=> ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_set_real_o(member_real(X2),B) ) ) ).

% subset_code(1)
tff(fact_3782_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),B)
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_set_nat_o(member_nat(X2),B) ) ) ).

% subset_code(1)
tff(fact_3783_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_int,Xsi2: list_int,A: fun_int_fun_int_assn,A5: fun_int_fun_int_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: int,Xi2: int] :
              ( aa_set_int_o(member_int(X3),set_int2(Xs4))
             => ( aa_set_int_o(member_int(Xi2),set_int2(Xsi2))
               => ( aa_int_assn(aa_int_fun_int_assn(A,X3),Xi2) = aa_int_assn(aa_int_fun_int_assn(A5,X3),Xi2) ) ) )
         => ( vEBT_L74593716426352029nt_int(A,Xs,Xsi) = vEBT_L74593716426352029nt_int(A5,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_3784_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_complex,Xsi2: list_complex,A: fun_in9020965360991744636x_assn,A5: fun_in9020965360991744636x_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: int,Xi2: complex] :
              ( aa_set_int_o(member_int(X3),set_int2(Xs4))
             => ( aa_set_complex_o(member_complex(Xi2),set_complex2(Xsi2))
               => ( aa_complex_assn(aa_int6135251106891545917x_assn(A,X3),Xi2) = aa_complex_assn(aa_int6135251106891545917x_assn(A5,X3),Xi2) ) ) )
         => ( vEBT_L6716599654974302751omplex(A,Xs,Xsi) = vEBT_L6716599654974302751omplex(A5,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_3785_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs4: list_complex,Xsi: list_int,Xsi2: list_int,A: fun_co6164383388265235324t_assn,A5: fun_co6164383388265235324t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: complex,Xi2: int] :
              ( aa_set_complex_o(member_complex(X3),set_complex2(Xs4))
             => ( aa_set_int_o(member_int(Xi2),set_int2(Xsi2))
               => ( aa_int_assn(aa_com2522482552906347837t_assn(A,X3),Xi2) = aa_int_assn(aa_com2522482552906347837t_assn(A5,X3),Xi2) ) ) )
         => ( vEBT_L134985006839036959ex_int(A,Xs,Xsi) = vEBT_L134985006839036959ex_int(A5,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_3786_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs4: list_complex,Xsi: list_complex,Xsi2: list_complex,A: fun_co6163879640491479806x_assn,A5: fun_co6163879640491479806x_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: complex,Xi2: complex] :
              ( aa_set_complex_o(member_complex(X3),set_complex2(Xs4))
             => ( aa_set_complex_o(member_complex(Xi2),set_complex2(Xsi2))
               => ( aa_complex_assn(aa_com5932308198457167295x_assn(A,X3),Xi2) = aa_complex_assn(aa_com5932308198457167295x_assn(A5,X3),Xi2) ) ) )
         => ( vEBT_L4260503343685368993omplex(A,Xs,Xsi) = vEBT_L4260503343685368993omplex(A5,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_3787_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: fun_in7755481285845205148T_assn,A5: fun_in7755481285845205148T_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: int,Xi2: vEBT_VEBT] :
              ( aa_set_int_o(member_int(X3),set_int2(Xs4))
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xi2),set_VEBT_VEBT2(Xsi2))
               => ( aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A,X3),Xi2) = aa_VEBT_VEBT_assn(aa_int878501045946686085T_assn(A5,X3),Xi2) ) ) )
         => ( vEBT_L1664421287176695555T_VEBT(A,Xs,Xsi) = vEBT_L1664421287176695555T_VEBT(A5,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_3788_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs4: list_complex,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: fun_co2031555705788908698T_assn,A5: fun_co2031555705788908698T_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: complex,Xi2: vEBT_VEBT] :
              ( aa_set_complex_o(member_complex(X3),set_complex2(Xs4))
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xi2),set_VEBT_VEBT2(Xsi2))
               => ( aa_VEBT_VEBT_assn(aa_com5451320767305428611T_assn(A,X3),Xi2) = aa_VEBT_VEBT_assn(aa_com5451320767305428611T_assn(A5,X3),Xi2) ) ) )
         => ( vEBT_L8524933119956041985T_VEBT(A,Xs,Xsi) = vEBT_L8524933119956041985T_VEBT(A5,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_3789_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_nat,Xsi2: list_nat,A: fun_int_fun_nat_assn,A5: fun_int_fun_nat_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: int,Xi2: nat] :
              ( aa_set_int_o(member_int(X3),set_int2(Xs4))
             => ( aa_set_nat_o(member_nat(Xi2),set_nat2(Xsi2))
               => ( aa_nat_assn(aa_int_fun_nat_assn(A,X3),Xi2) = aa_nat_assn(aa_int_fun_nat_assn(A5,X3),Xi2) ) ) )
         => ( vEBT_L77084186935402305nt_nat(A,Xs,Xsi) = vEBT_L77084186935402305nt_nat(A5,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_3790_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs4: list_complex,Xsi: list_nat,Xsi2: list_nat,A: fun_co1748589094918083360t_assn,A5: fun_co1748589094918083360t_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: complex,Xi2: nat] :
              ( aa_set_complex_o(member_complex(X3),set_complex2(Xs4))
             => ( aa_set_nat_o(member_nat(Xi2),set_nat2(Xsi2))
               => ( aa_nat_assn(aa_com1546947087700170465t_assn(A,X3),Xi2) = aa_nat_assn(aa_com1546947087700170465t_assn(A5,X3),Xi2) ) ) )
         => ( vEBT_L137475477348087235ex_nat(A,Xs,Xsi) = vEBT_L137475477348087235ex_nat(A5,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_3791_list__assn__cong,axiom,
    ! [Xs: list_int,Xs4: list_int,Xsi: list_real,Xsi2: list_real,A: fun_in6446958447386763770l_assn,A5: fun_in6446958447386763770l_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: int,Xi2: real] :
              ( aa_set_int_o(member_int(X3),set_int2(Xs4))
             => ( aa_set_real_o(member_real(Xi2),set_real2(Xsi2))
               => ( aa_real_assn(aa_int_fun_real_assn(A,X3),Xi2) = aa_real_assn(aa_int_fun_real_assn(A5,X3),Xi2) ) ) )
         => ( vEBT_L8288995350762215837t_real(A,Xs,Xsi) = vEBT_L8288995350762215837t_real(A5,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_3792_list__assn__cong,axiom,
    ! [Xs: list_complex,Xs4: list_complex,Xsi: list_real,Xsi2: list_real,A: fun_co8000059489268409724l_assn,A5: fun_co8000059489268409724l_assn] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X3: complex,Xi2: real] :
              ( aa_set_complex_o(member_complex(X3),set_complex2(Xs4))
             => ( aa_set_real_o(member_real(Xi2),set_real2(Xsi2))
               => ( aa_real_assn(aa_com7355656519465148733l_assn(A,X3),Xi2) = aa_real_assn(aa_com7355656519465148733l_assn(A5,X3),Xi2) ) ) )
         => ( vEBT_L2479436891206192927x_real(A,Xs,Xsi) = vEBT_L2479436891206192927x_real(A5,Xs4,Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_3793_set__update__subsetI,axiom,
    ! [Xs: list_int,A: set_int,Xc: int,I: nat] :
      ( aa_set_int_o(ord_less_eq_set_int(set_int2(Xs)),A)
     => ( aa_set_int_o(member_int(Xc),A)
       => aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xc))),A) ) ) ).

% set_update_subsetI
tff(fact_3794_set__update__subsetI,axiom,
    ! [Xs: list_complex,A: set_complex,Xc: complex,I: nat] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Xs)),A)
     => ( aa_set_complex_o(member_complex(Xc),A)
       => aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(list_update_complex(Xs,I,Xc))),A) ) ) ).

% set_update_subsetI
tff(fact_3795_set__update__subsetI,axiom,
    ! [Xs: list_real,A: set_real,Xc: real,I: nat] :
      ( aa_set_real_o(ord_less_eq_set_real(set_real2(Xs)),A)
     => ( aa_set_real_o(member_real(Xc),A)
       => aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xc))),A) ) ) ).

% set_update_subsetI
tff(fact_3796_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBTi,A: set_VEBT_VEBTi,Xc: vEBT_VEBTi,I: nat] :
      ( ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(Xs),A)
     => ( member_VEBT_VEBTi(Xc,A)
       => ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,Xc)),A) ) ) ).

% set_update_subsetI
tff(fact_3797_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A: set_VEBT_VEBT,Xc: vEBT_VEBT,I: nat] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Xs)),A)
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),A)
       => aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xc))),A) ) ) ).

% set_update_subsetI
tff(fact_3798_set__update__subsetI,axiom,
    ! [Xs: list_nat,A: set_nat,Xc: nat,I: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Xs)),A)
     => ( aa_set_nat_o(member_nat(Xc),A)
       => aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,Xc))),A) ) ) ).

% set_update_subsetI
tff(fact_3799_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw: $o,Nb: nat] : vEBT_vebt_succ(aa_o_VEBT_VEBT(vEBT_Leaf((Uv2)),(Uw)),aa_nat_nat(suc,Nb)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_3800_vebt__insert_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xc: nat] :
      aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),Xc) = $ite(
        Xc = zero_zero_nat,
        aa_o_VEBT_VEBT(vEBT_Leaf($true),(B2)),
        $ite(Xc = one_one_nat,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),$true),aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))) ) ).

% vebt_insert.simps(1)
tff(fact_3801_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] : vEBT_vebt_pred(aa_o_VEBT_VEBT(vEBT_Leaf((Uu)),(Uv2)),zero_zero_nat) = none_nat ).

% vebt_pred.simps(1)
tff(fact_3802_vebt__buildup_Osimps_I1_J,axiom,
    aa_nat_VEBT_VEBT(vEBT_vebt_buildup,zero_zero_nat) = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) ).

% vebt_buildup.simps(1)
tff(fact_3803_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xc: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),X3)
     => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw2),Ux)
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_3804_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_3805_length__pos__if__in__set,axiom,
    ! [Xc: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xc),set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3806_length__pos__if__in__set,axiom,
    ! [Xc: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(Xc),set_complex2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3807_length__pos__if__in__set,axiom,
    ! [Xc: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3808_length__pos__if__in__set,axiom,
    ! [Xc: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xc),set_real2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_real(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3809_length__pos__if__in__set,axiom,
    ! [Xc: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((Xc)),set_o2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_o(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3810_length__pos__if__in__set,axiom,
    ! [Xc: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xc),set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3811_length__pos__if__in__set,axiom,
    ! [Xc: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( member_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s7982070591426661849_VEBTi(Xs)) ) ).

% length_pos_if_in_set
tff(fact_3812_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X2) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3813_all__set__conv__all__nth,axiom,
    ! [Xs: list_real,P: fun_real_o] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Xs))
         => aa_real_o(P,X2) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3814_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),set_o2(Xs))
         => aa_o_o(P,(X2)) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3815_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Xs))
         => aa_nat_o(P,X2) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3816_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( ! [X2: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(Xs))
         => aa_VEBT_VEBTi_o(P,X2) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,I2)) ) ) ).

% all_set_conv_all_nth
tff(fact_3817_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xc: int] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I5)) )
     => ( aa_set_int_o(member_int(Xc),set_int2(Xs))
       => aa_int_o(P,Xc) ) ) ).

% all_nth_imp_all_set
tff(fact_3818_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Xc: complex] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s3451745648224563538omplex(Xs))
         => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),I5)) )
     => ( aa_set_complex_o(member_complex(Xc),set_complex2(Xs))
       => aa_complex_o(P,Xc) ) ) ).

% all_nth_imp_all_set
tff(fact_3819_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xc: vEBT_VEBT] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I5)) )
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,Xc) ) ) ).

% all_nth_imp_all_set
tff(fact_3820_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,Xc: real] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I5)) )
     => ( aa_set_real_o(member_real(Xc),set_real2(Xs))
       => aa_real_o(P,Xc) ) ) ).

% all_nth_imp_all_set
tff(fact_3821_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,Xc: $o] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I5)) )
     => ( aa_set_o_o(member_o((Xc)),set_o2(Xs))
       => aa_o_o(P,(Xc)) ) ) ).

% all_nth_imp_all_set
tff(fact_3822_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Xc: nat] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I5)) )
     => ( aa_set_nat_o(member_nat(Xc),set_nat2(Xs))
       => aa_nat_o(P,Xc) ) ) ).

% all_nth_imp_all_set
tff(fact_3823_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o,Xc: vEBT_VEBTi] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_s7982070591426661849_VEBTi(Xs))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,I5)) )
     => ( member_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(Xs))
       => aa_VEBT_VEBTi_o(P,Xc) ) ) ).

% all_nth_imp_all_set
tff(fact_3824_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(L))
         => aa_VEBT_VEBT_o(P,X2) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(L))
         => aa_VEBT_VEBT_o(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_3825_all__set__conv__nth,axiom,
    ! [L: list_real,P: fun_real_o] :
      ( ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(L))
         => aa_real_o(P,X2) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(L))
         => aa_real_o(P,aa_nat_real(nth_real(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_3826_all__set__conv__nth,axiom,
    ! [L: list_o,P: fun_o_o] :
      ( ! [X2: $o] :
          ( aa_set_o_o(member_o((X2)),set_o2(L))
         => aa_o_o(P,(X2)) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(L))
         => aa_o_o(P,aa_nat_o(nth_o(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_3827_all__set__conv__nth,axiom,
    ! [L: list_nat,P: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(L))
         => aa_nat_o(P,X2) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(L))
         => aa_nat_o(P,aa_nat_nat(nth_nat(L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_3828_all__set__conv__nth,axiom,
    ! [L: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( ! [X2: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X2,set_VEBT_VEBTi2(L))
         => aa_VEBT_VEBTi_o(P,X2) )
    <=> ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(L))
         => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(L,I2)) ) ) ).

% all_set_conv_nth
tff(fact_3829_in__set__conv__nth,axiom,
    ! [Xc: int,Xs: list_int] :
      ( aa_set_int_o(member_int(Xc),set_int2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
          & ( aa_nat_int(nth_int(Xs),I2) = Xc ) ) ) ).

% in_set_conv_nth
tff(fact_3830_in__set__conv__nth,axiom,
    ! [Xc: complex,Xs: list_complex] :
      ( aa_set_complex_o(member_complex(Xc),set_complex2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3451745648224563538omplex(Xs))
          & ( aa_nat_complex(nth_complex(Xs),I2) = Xc ) ) ) ).

% in_set_conv_nth
tff(fact_3831_in__set__conv__nth,axiom,
    ! [Xc: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),set_VEBT_VEBT2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
          & ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I2) = Xc ) ) ) ).

% in_set_conv_nth
tff(fact_3832_in__set__conv__nth,axiom,
    ! [Xc: real,Xs: list_real] :
      ( aa_set_real_o(member_real(Xc),set_real2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I2) = Xc ) ) ) ).

% in_set_conv_nth
tff(fact_3833_in__set__conv__nth,axiom,
    ! [Xc: $o,Xs: list_o] :
      ( aa_set_o_o(member_o((Xc)),set_o2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
          & ( aa_nat_o(nth_o(Xs),I2)
          <=> (Xc) ) ) ) ).

% in_set_conv_nth
tff(fact_3834_in__set__conv__nth,axiom,
    ! [Xc: nat,Xs: list_nat] :
      ( aa_set_nat_o(member_nat(Xc),set_nat2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
          & ( aa_nat_nat(nth_nat(Xs),I2) = Xc ) ) ) ).

% in_set_conv_nth
tff(fact_3835_in__set__conv__nth,axiom,
    ! [Xc: vEBT_VEBTi,Xs: list_VEBT_VEBTi] :
      ( member_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7982070591426661849_VEBTi(Xs))
          & ( nth_VEBT_VEBTi(Xs,I2) = Xc ) ) ) ).

% in_set_conv_nth
tff(fact_3836_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( ! [X3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs))
           => aa_VEBT_VEBT_o(P,X3) )
       => aa_VEBT_VEBT_o(P,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_3837_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_real,P: fun_real_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),set_real2(Xs))
           => aa_real_o(P,X3) )
       => aa_real_o(P,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_3838_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_o,P: fun_o_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),set_o2(Xs))
           => aa_o_o(P,(X3)) )
       => aa_o_o(P,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_3839_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
           => aa_nat_o(P,X3) )
       => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_3840_list__ball__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,P: fun_VEBT_VEBTi_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => ( ! [X3: vEBT_VEBTi] :
            ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(Xs))
           => aa_VEBT_VEBTi_o(P,X3) )
       => aa_VEBT_VEBTi_o(P,nth_VEBT_VEBTi(Xs,Nb)) ) ) ).

% list_ball_nth
tff(fact_3841_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int(aa_nat_int(nth_int(Xs),Nb)),set_int2(Xs)) ) ).

% nth_mem
tff(fact_3842_nth__mem,axiom,
    ! [Nb: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(aa_nat_complex(nth_complex(Xs),Nb)),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_3843_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)),set_VEBT_VEBT2(Xs)) ) ).

% nth_mem
tff(fact_3844_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real(aa_nat_real(nth_real(Xs),Nb)),set_real2(Xs)) ) ).

% nth_mem
tff(fact_3845_nth__mem,axiom,
    ! [Nb: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => aa_set_o_o(member_o(aa_nat_o(nth_o(Xs),Nb)),set_o2(Xs)) ) ).

% nth_mem
tff(fact_3846_nth__mem,axiom,
    ! [Nb: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(aa_nat_nat(nth_nat(Xs),Nb)),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_3847_nth__mem,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => member_VEBT_VEBTi(nth_VEBT_VEBTi(Xs,Nb),set_VEBT_VEBTi2(Xs)) ) ).

% nth_mem
tff(fact_3848_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [Xc: vEBT_VEBT] :
      ( ( Xc != aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) )
     => ( ! [Uv: $o] : Xc != aa_o_VEBT_VEBT(vEBT_Leaf($true),(Uv))
       => ( ! [Uu2: $o] : Xc != aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),$true)
         => ( ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux),Uy)
           => ~ ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_3849_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_int,Xc: int,Ya: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_set_int_o(member_int(Xc),set_int2(list_update_int(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ! [Y4: int] : aa_set_int_o(member_int(Xc),set_int2(list_update_int(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3850_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_complex,Xc: complex,Ya: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3451745648224563538omplex(L))
     => ( aa_set_complex_o(member_complex(Xc),set_complex2(list_update_complex(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ! [Y4: complex] : aa_set_complex_o(member_complex(Xc),set_complex2(list_update_complex(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3851_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xc: vEBT_VEBT,Ya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ! [Y4: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3852_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_real,Xc: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_set_real_o(member_real(Xc),set_real2(list_update_real(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ! [Y4: real] : aa_set_real_o(member_real(Xc),set_real2(list_update_real(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3853_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_o,Xc: $o,Ya: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( aa_set_o_o(member_o((Xc)),set_o2(list_update_o(L,I,(Ya))))
      <=> ( ( (Xc)
          <=> (Ya) )
          | ! [Y4: $o] : aa_set_o_o(member_o((Xc)),set_o2(list_update_o(L,I,(Y4)))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3854_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_set_nat_o(member_nat(Xc),set_nat2(list_update_nat(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ! [Y4: nat] : aa_set_nat_o(member_nat(Xc),set_nat2(list_update_nat(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3855_in__set__upd__eq__aux,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xc: vEBT_VEBTi,Ya: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( member_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ! [Y4: vEBT_VEBTi] : member_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y4))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_3856_in__set__upd__cases,axiom,
    ! [Xc: int,L: list_int,I: nat,Ya: int] :
      ( aa_set_int_o(member_int(Xc),set_int2(list_update_int(L,I,Ya)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
         => ( Xc != Ya ) )
       => aa_set_int_o(member_int(Xc),set_int2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3857_in__set__upd__cases,axiom,
    ! [Xc: complex,L: list_complex,I: nat,Ya: complex] :
      ( aa_set_complex_o(member_complex(Xc),set_complex2(list_update_complex(L,I,Ya)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3451745648224563538omplex(L))
         => ( Xc != Ya ) )
       => aa_set_complex_o(member_complex(Xc),set_complex2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3858_in__set__upd__cases,axiom,
    ! [Xc: vEBT_VEBT,L: list_VEBT_VEBT,I: nat,Ya: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Ya)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
         => ( Xc != Ya ) )
       => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),set_VEBT_VEBT2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3859_in__set__upd__cases,axiom,
    ! [Xc: real,L: list_real,I: nat,Ya: real] :
      ( aa_set_real_o(member_real(Xc),set_real2(list_update_real(L,I,Ya)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
         => ( Xc != Ya ) )
       => aa_set_real_o(member_real(Xc),set_real2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3860_in__set__upd__cases,axiom,
    ! [Xc: $o,L: list_o,I: nat,Ya: $o] :
      ( aa_set_o_o(member_o((Xc)),set_o2(list_update_o(L,I,(Ya))))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
         => ( (Xc)
          <=> ~ (Ya) ) )
       => aa_set_o_o(member_o((Xc)),set_o2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3861_in__set__upd__cases,axiom,
    ! [Xc: nat,L: list_nat,I: nat,Ya: nat] :
      ( aa_set_nat_o(member_nat(Xc),set_nat2(list_update_nat(L,I,Ya)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
         => ( Xc != Ya ) )
       => aa_set_nat_o(member_nat(Xc),set_nat2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3862_in__set__upd__cases,axiom,
    ! [Xc: vEBT_VEBTi,L: list_VEBT_VEBTi,I: nat,Ya: vEBT_VEBTi] :
      ( member_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Ya)))
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
         => ( Xc != Ya ) )
       => member_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(L)) ) ) ).

% in_set_upd_cases
tff(fact_3863_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_int,Xc: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => aa_set_int_o(member_int(Xc),set_int2(list_update_int(Xs,Nb,Xc))) ) ).

% set_update_memI
tff(fact_3864_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_complex,Xc: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => aa_set_complex_o(member_complex(Xc),set_complex2(list_update_complex(Xs,Nb,Xc))) ) ).

% set_update_memI
tff(fact_3865_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Xc: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,Nb,Xc))) ) ).

% set_update_memI
tff(fact_3866_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_real,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => aa_set_real_o(member_real(Xc),set_real2(list_update_real(Xs,Nb,Xc))) ) ).

% set_update_memI
tff(fact_3867_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_o,Xc: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => aa_set_o_o(member_o((Xc)),set_o2(list_update_o(Xs,Nb,(Xc)))) ) ).

% set_update_memI
tff(fact_3868_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_nat(Xs))
     => aa_set_nat_o(member_nat(Xc),set_nat2(list_update_nat(Xs,Nb,Xc))) ) ).

% set_update_memI
tff(fact_3869_set__update__memI,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBTi,Xc: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s7982070591426661849_VEBTi(Xs))
     => member_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,Nb,Xc))) ) ).

% set_update_memI
tff(fact_3870_in__set__upd__eq,axiom,
    ! [I: nat,L: list_int,Xc: int,Ya: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( aa_set_int_o(member_int(Xc),set_int2(list_update_int(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ( aa_set_int_o(member_int(Xc),set_int2(L))
            & ! [Y4: int] : aa_set_int_o(member_int(Xc),set_int2(list_update_int(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3871_in__set__upd__eq,axiom,
    ! [I: nat,L: list_complex,Xc: complex,Ya: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s3451745648224563538omplex(L))
     => ( aa_set_complex_o(member_complex(Xc),set_complex2(list_update_complex(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ( aa_set_complex_o(member_complex(Xc),set_complex2(L))
            & ! [Y4: complex] : aa_set_complex_o(member_complex(Xc),set_complex2(list_update_complex(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3872_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xc: vEBT_VEBT,Ya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),set_VEBT_VEBT2(L))
            & ! [Y4: vEBT_VEBT] : aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Xc),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3873_in__set__upd__eq,axiom,
    ! [I: nat,L: list_real,Xc: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( aa_set_real_o(member_real(Xc),set_real2(list_update_real(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ( aa_set_real_o(member_real(Xc),set_real2(L))
            & ! [Y4: real] : aa_set_real_o(member_real(Xc),set_real2(list_update_real(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3874_in__set__upd__eq,axiom,
    ! [I: nat,L: list_o,Xc: $o,Ya: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( aa_set_o_o(member_o((Xc)),set_o2(list_update_o(L,I,(Ya))))
      <=> ( ( (Xc)
          <=> (Ya) )
          | ( aa_set_o_o(member_o((Xc)),set_o2(L))
            & ! [Y4: $o] : aa_set_o_o(member_o((Xc)),set_o2(list_update_o(L,I,(Y4)))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3875_in__set__upd__eq,axiom,
    ! [I: nat,L: list_nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( aa_set_nat_o(member_nat(Xc),set_nat2(list_update_nat(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ( aa_set_nat_o(member_nat(Xc),set_nat2(L))
            & ! [Y4: nat] : aa_set_nat_o(member_nat(Xc),set_nat2(list_update_nat(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3876_in__set__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xc: vEBT_VEBTi,Ya: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( member_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Ya)))
      <=> ( ( Xc = Ya )
          | ( member_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(L))
            & ! [Y4: vEBT_VEBTi] : member_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Y4))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_3877_vebt__buildup_Osimps_I2_J,axiom,
    aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,zero_zero_nat)) = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) ).

% vebt_buildup.simps(2)
tff(fact_3878_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,Xc: int] : aa_set_int_o(ord_less_eq_set_int(set_int2(list_update_int(Xs,I,Xc))),insert_int(Xc,set_int2(Xs))) ).

% set_update_subset_insert
tff(fact_3879_set__update__subset__insert,axiom,
    ! [Xs: list_o,I: nat,Xc: $o] : aa_set_o_o(ord_less_eq_set_o(set_o2(list_update_o(Xs,I,(Xc)))),insert_o((Xc),set_o2(Xs))) ).

% set_update_subset_insert
tff(fact_3880_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,Xc: real] : aa_set_real_o(ord_less_eq_set_real(set_real2(list_update_real(Xs,I,Xc))),insert_real(Xc,set_real2(Xs))) ).

% set_update_subset_insert
tff(fact_3881_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBTi,I: nat,Xc: vEBT_VEBTi] : ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(Xs,I,Xc)),insert_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(Xs))) ).

% set_update_subset_insert
tff(fact_3882_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,Xc: vEBT_VEBT] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(Xs,I,Xc))),insert_VEBT_VEBT(Xc,set_VEBT_VEBT2(Xs))) ).

% set_update_subset_insert
tff(fact_3883_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,Xc: nat] : aa_set_nat_o(ord_less_eq_set_nat(set_nat2(list_update_nat(Xs,I,Xc))),insert_nat(Xc,set_nat2(Xs))) ).

% set_update_subset_insert
tff(fact_3884_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Nb: nat] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_3885_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Nb: nat] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_3886_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),zero_zero_nat) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_3887_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),zero_zero_nat) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
tff(fact_3888_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] :
      vEBT_vebt_pred(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(Uw)),aa_nat_nat(suc,zero_zero_nat)) = $ite((A2),some_nat(zero_zero_nat),none_nat) ).

% vebt_pred.simps(2)
tff(fact_3889_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] :
      vEBT_vebt_succ(aa_o_VEBT_VEBT(vEBT_Leaf((Uu)),(B2)),zero_zero_nat) = $ite((B2),some_nat(one_one_nat),none_nat) ).

% vebt_succ.simps(1)
tff(fact_3890_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xc: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] : Xc != aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
     => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
       => ~ ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_3891_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] : aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
tff(fact_3892_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] : aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
tff(fact_3893_vebt__pred_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Vaa: nat] :
      vEBT_vebt_pred(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = $ite(
        (B2),
        some_nat(one_one_nat),
        $ite((A2),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_pred.simps(3)
tff(fact_3894_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_int,Xc: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_int(L))
     => ( insert_int(aa_nat_int(nth_int(L),I),set_int2(list_update_int(L,I,Xc))) = insert_int(Xc,set_int2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_3895_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,Xc: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
     => ( insert_VEBT_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I),set_VEBT_VEBT2(list_u1324408373059187874T_VEBT(L,I,Xc))) = insert_VEBT_VEBT(Xc,set_VEBT_VEBT2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_3896_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_real,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(L))
     => ( insert_real(aa_nat_real(nth_real(L),I),set_real2(list_update_real(L,I,Xc))) = insert_real(Xc,set_real2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_3897_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_o,Xc: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(L))
     => ( insert_o(aa_nat_o(nth_o(L),I),set_o2(list_update_o(L,I,(Xc)))) = insert_o((Xc),set_o2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_3898_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(L))
     => ( insert_nat(aa_nat_nat(nth_nat(L),I),set_nat2(list_update_nat(L,I,Xc))) = insert_nat(Xc,set_nat2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_3899_insert__swap__set__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBTi,Xc: vEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(L))
     => ( insert_VEBT_VEBTi(nth_VEBT_VEBTi(L,I),set_VEBT_VEBTi2(list_u6098035379799741383_VEBTi(L,I,Xc))) = insert_VEBT_VEBTi(Xc,set_VEBT_VEBTi2(L)) ) ) ).

% insert_swap_set_eq
tff(fact_3900_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [Xc: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),X3)
     => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2),X3)
       => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz),X3)
         => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc),X3)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),X3) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_3901_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [Xc: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),X3)
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),X3)
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),X3)
         => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary),X3)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),X3) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_3902_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [Xc: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B3: $o] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(B3)),zero_zero_nat)
     => ( ! [Uv: $o,Uw2: $o,N: nat] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uv)),(Uw2)),aa_nat_nat(suc,N))
       => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz),Va)
         => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd),Ve)
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Vi2)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),X3) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_3903_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [Xc: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv: $o] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)),zero_zero_nat)
     => ( ! [A3: $o,Uw2: $o] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(Uw2)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: $o,B3: $o,Va2: nat] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))
         => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va),Vb)
           => ( ! [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd),Ve),Vf)
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Vj2)
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),X3) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_3904_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [Xc: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),zero_zero_nat)
     => ( ! [A3: $o,B3: $o] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,zero_zero_nat))
       => ( ! [A3: $o,B3: $o,N: nat] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary),Uu2)
           => ( ! [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList2),Summary),X3)
             => ( ! [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary),X3)
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),X3) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_3905_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xc: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv: $o,Uw2: nat] : Xc != produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)),Uw2)
     => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Uz)
       => ( ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb),X3)
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList2),Vc),X3)
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] : Xc != produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3906_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A1),A22)
    <=> ( ( ? [A6: $o,B6: $o] : A1 = aa_o_VEBT_VEBT(vEBT_Leaf((A6)),(B6))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList3: list_VEBT_VEBT,N6: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary2) )
            & ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),N6)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),N6) )
            & ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_1)
            & ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_1) ) )
        | ? [TreeList3: list_VEBT_VEBT,N6: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList3),Summary2) )
            & ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),aa_nat_nat(suc,N6))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N6)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),aa_nat_nat(suc,N6)) )
            & ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),X_1)
            & ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList3))
               => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_1) ) )
        | ? [TreeList3: list_VEBT_VEBT,N6: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList3),Summary2) )
            & ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),N6)
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),N6) )
            & ! [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6))
               => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I2)),X_1)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I2) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_1) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N6) = I2 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N6)) )
                    & ! [X2: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N6) = I2 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N6)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X2)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma2) ) ) ) ) ) )
        | ? [TreeList3: list_VEBT_VEBT,N6: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList3),Summary2) )
            & ! [X2: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList3))
               => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X2),N6) )
            & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary2),aa_nat_nat(suc,N6))
            & ( size_s6755466524823107622T_VEBT(TreeList3) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N6)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N6),aa_nat_nat(suc,N6)) )
            & ! [I2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N6)))
               => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I2)),X_1)
                <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary2),I2) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X2: vEBT_VEBT] :
                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(TreeList3))
                 => ~ ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X2),X_1) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N6)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N6) = I2 )
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N6)) )
                    & ! [X2: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X2),N6) = I2 )
                          & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList3),I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X2),N6)) )
                       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi2),X2)
                          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3907_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,A1),A22)
     => ( ( ? [A3: $o,B3: $o] : A1 = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( A22 != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary) )
             => ( ( A22 = Deg2 )
               => ( ! [X4: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),N) )
                 => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M4)
                   => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                     => ( ( M4 = N )
                       => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                         => ( ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_13)
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                                 => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_13) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary) )
               => ( ( A22 = Deg2 )
                 => ( ! [X4: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),N) )
                   => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M4)
                     => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                       => ( ( M4 = aa_nat_nat(suc,N) )
                         => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                           => ( ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),X_13)
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                                   => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_13) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg2,TreeList2),Summary) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X4: vEBT_VEBT] :
                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),N) )
                     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M4)
                       => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                         => ( ( M4 = N )
                           => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                             => ( ! [I6: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I6),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4))
                                   => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I6)),X_1)
                                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I6) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X4: vEBT_VEBT] :
                                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                                       => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_13) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2))
                                     => ~ ( ( Mi != Ma )
                                         => ! [I6: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I6),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I6 )
                                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I6)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                & ! [X4: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N) = I6 )
                                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I6)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X4)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M4: nat,Deg2: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg2,TreeList2),Summary) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X4: vEBT_VEBT] :
                            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X4),N) )
                       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summary),M4)
                         => ( ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4) )
                           => ( ( M4 = aa_nat_nat(suc,N) )
                             => ( ( Deg2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),M4) )
                               => ( ! [I6: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I6),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4))
                                     => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I6)),X_1)
                                      <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summary),I6) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X4: vEBT_VEBT] :
                                          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),set_VEBT_VEBT2(TreeList2))
                                         => ~ ? [X_13: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X4),X_13) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg2))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I6: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I6),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M4))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N) = I6 )
                                                   => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I6)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N)) )
                                                  & ! [X4: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N) = I6 )
                                                        & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),I6)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X4)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3908_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Nb )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),X_12)
               => ( ! [X3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_12) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_3909_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,Nb) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),X_12)
               => ( ! [X3: vEBT_VEBT] :
                      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_12) )
                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya)),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_3910_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,Xc: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2) ).

% vebt_insert.simps(2)
tff(fact_3911_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xc),Ya)
    <=> ( aa_set_nat_o(member_nat(Ya),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
        & ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),X2)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),X2) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_3912_is__pred__in__set__def,axiom,
    ! [Xs: set_nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),Xc),Ya)
    <=> ( aa_set_nat_o(member_nat(Ya),Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc)
        & ! [X2: nat] :
            ( aa_set_nat_o(member_nat(X2),Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),Xc)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Ya) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_3913_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,Xc: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2) ).

% vebt_insert.simps(3)
tff(fact_3914_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Vaa: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2),Vaa) = none_nat ).

% vebt_succ.simps(3)
tff(fact_3915_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz2: list_VEBT_VEBT,Vaa: vEBT_VEBT,Vb2: nat] : vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Vaa),Vb2) = none_nat ).

% vebt_pred.simps(4)
tff(fact_3916_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat,Mia: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = Nb )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ! [I5: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),I5)),X_1)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),I5) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_12) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                     => ( ( ( Mia != Maa )
                         => ! [I5: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I5 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Nb) = I5 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_3917_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat,Mia: nat,Maa: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
         => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),Nb) )
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Summarya),M)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) )
         => ( ( M = aa_nat_nat(suc,Nb) )
           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M) )
             => ( ! [I5: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                   => ( ? [X_1: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),I5)),X_1)
                    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Summarya),I5) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,X3),X_12) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                     => ( ( ( Mia != Maa )
                         => ! [I5: nat] :
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I5 )
                                 => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Maa),Nb)) )
                                & ! [X3: nat] :
                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X3),Nb) = I5 )
                                      & aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),I5)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),X3)
                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Maa) ) ) ) ) )
                       => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_3918_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd2),Ve2) = none_nat ).

% vebt_succ.simps(4)
tff(fact_3919_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd2),Ve2),Vf2) = none_nat ).

% vebt_pred.simps(5)
tff(fact_3920_vebt__maxt_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: option_nat] :
      ( ( vEBT_vebt_maxt(Xc) = Ya )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => ( Ya != $ite(
                  (B3),
                  some_nat(one_one_nat),
                  $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
           => ( Ya != none_nat ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)
               => ( Ya != some_nat(Ma) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_3921_vebt__mint_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: option_nat] :
      ( ( vEBT_vebt_mint(Xc) = Ya )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => ( Ya != $ite(
                  (A3),
                  some_nat(zero_zero_nat),
                  $ite((B3),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
           => ( Ya != none_nat ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)
               => ( Ya != some_nat(Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_3922_vebt__maxt_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_maxt(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))) = $ite(
        (B2),
        some_nat(one_one_nat),
        $ite((A2),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_maxt.simps(1)
tff(fact_3923_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_set_nat_o(ord_less_eq_set_nat(vEBT_VEBT_set_vebt(Ta)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ) ).

% inrange
tff(fact_3924_greater__shift,axiom,
    ! [Ya: nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(Xc)),some_nat(Ya)) ) ).

% greater_shift
tff(fact_3925_less__shift,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
    <=> aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(Xc)),some_nat(Ya)) ) ).

% less_shift
tff(fact_3926_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(L),I)
        & aa_set_nat_o(ord_less_eq_set_nat(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3927_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( aa_set_rat_o(member_rat(I),set_or633870826150836451st_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3928_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( aa_set_num_o(member_num(I),set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3929_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( aa_set_nat_o(member_nat(I),aa_nat_set_nat(set_or1269000886237332187st_nat(L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3930_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( aa_set_int_o(member_int(I),set_or1266510415728281911st_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3931_atLeastAtMost__iff,axiom,
    ! [I: code_integer,L: code_integer,U: code_integer] :
      ( aa_set5643387332196575111eger_o(member_Code_integer(I),set_or189985376899183464nteger(L,U))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),I)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3932_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( aa_set_real_o(member_real(I),set_or1222579329274155063t_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_3933_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_set_nat_o(ord_less_eq_set_nat(L),H)
          & ~ aa_set_nat_o(ord_less_eq_set_nat(L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3934_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L2: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
          & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3935_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L2: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),H)
          & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3936_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L2: nat,H2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(L),H) = aa_nat_set_nat(set_or1269000886237332187st_nat(L2),H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3937_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L2: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
          & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3938_Icc__eq__Icc,axiom,
    ! [L: code_integer,H: code_integer,L2: code_integer,H2: code_integer] :
      ( ( set_or189985376899183464nteger(L,H) = set_or189985376899183464nteger(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L),H)
          & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3939_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L2: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L2,H2) )
    <=> ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
          & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L2),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_3940_atLeastatMost__empty__iff2,axiom,
    ! [A2: $o,B2: $o] :
      ( ( bot_bot_set_o = set_or8904488021354931149Most_o((A2),(B2)) )
    <=> ~ aa_o_o(ord_less_eq_o((A2)),(B2)) ) ).

% atLeastatMost_empty_iff2
tff(fact_3941_atLeastatMost__empty__iff2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A2,B2) )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3942_atLeastatMost__empty__iff2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A2,B2) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3943_atLeastatMost__empty__iff2,axiom,
    ! [A2: num,B2: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A2,B2) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3944_atLeastatMost__empty__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A2),B2) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3945_atLeastatMost__empty__iff2,axiom,
    ! [A2: int,B2: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A2,B2) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3946_atLeastatMost__empty__iff2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( bot_bo3990330152332043303nteger = set_or189985376899183464nteger(A2,B2) )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3947_atLeastatMost__empty__iff2,axiom,
    ! [A2: real,B2: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A2,B2) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_3948_atLeastatMost__empty__iff,axiom,
    ! [A2: $o,B2: $o] :
      ( ( set_or8904488021354931149Most_o((A2),(B2)) = bot_bot_set_o )
    <=> ~ aa_o_o(ord_less_eq_o((A2)),(B2)) ) ).

% atLeastatMost_empty_iff
tff(fact_3949_atLeastatMost__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( set_or4548717258645045905et_nat(A2,B2) = bot_bot_set_set_nat )
    <=> ~ aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3950_atLeastatMost__empty__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( set_or633870826150836451st_rat(A2,B2) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3951_atLeastatMost__empty__iff,axiom,
    ! [A2: num,B2: num] :
      ( ( set_or7049704709247886629st_num(A2,B2) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3952_atLeastatMost__empty__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A2),B2) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3953_atLeastatMost__empty__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( set_or1266510415728281911st_int(A2,B2) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3954_atLeastatMost__empty__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( set_or189985376899183464nteger(A2,B2) = bot_bo3990330152332043303nteger )
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3955_atLeastatMost__empty__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( set_or1222579329274155063t_real(A2,B2) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_3956_atLeastatMost__empty,axiom,
    ! [B2: $o,A2: $o] :
      ( aa_o_o(ord_less_o((B2)),(A2))
     => ( set_or8904488021354931149Most_o((A2),(B2)) = bot_bot_set_o ) ) ).

% atLeastatMost_empty
tff(fact_3957_atLeastatMost__empty,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( set_or633870826150836451st_rat(A2,B2) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_3958_atLeastatMost__empty,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
     => ( set_or7049704709247886629st_num(A2,B2) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_3959_atLeastatMost__empty,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A2),B2) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_3960_atLeastatMost__empty,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ( set_or1266510415728281911st_int(A2,B2) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_3961_atLeastatMost__empty,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),A2)
     => ( set_or189985376899183464nteger(A2,B2) = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
tff(fact_3962_atLeastatMost__empty,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( set_or1222579329274155063t_real(A2,B2) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_3963_atLeastatMost__subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat,D2: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A2,B2),set_or4548717258645045905et_nat(C2,D2))
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        | ( aa_set_nat_o(ord_less_eq_set_nat(C2),A2)
          & aa_set_nat_o(ord_less_eq_set_nat(B2),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3964_atLeastatMost__subset__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(A2,B2)),set_or633870826150836451st_rat(C2,D2))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
        | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3965_atLeastatMost__subset__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(A2,B2)),set_or7049704709247886629st_num(C2,D2))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
        | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A2)
          & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3966_atLeastatMost__subset__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A2),B2)),aa_nat_set_nat(set_or1269000886237332187st_nat(C2),D2))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3967_atLeastatMost__subset__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(A2,B2)),set_or1266510415728281911st_int(C2,D2))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3968_atLeastatMost__subset__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or189985376899183464nteger(A2,B2)),set_or189985376899183464nteger(C2,D2))
    <=> ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
        | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3969_atLeastatMost__subset__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or1222579329274155063t_real(A2,B2)),set_or1222579329274155063t_real(C2,D2))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
        | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),D2) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_3970_atLeastAtMost__singleton,axiom,
    ! [A2: $o] : set_or8904488021354931149Most_o((A2),(A2)) = insert_o((A2),bot_bot_set_o) ).

% atLeastAtMost_singleton
tff(fact_3971_atLeastAtMost__singleton,axiom,
    ! [A2: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(A2),A2) = insert_nat(A2,bot_bot_set_nat) ).

% atLeastAtMost_singleton
tff(fact_3972_atLeastAtMost__singleton,axiom,
    ! [A2: int] : set_or1266510415728281911st_int(A2,A2) = insert_int(A2,bot_bot_set_int) ).

% atLeastAtMost_singleton
tff(fact_3973_atLeastAtMost__singleton,axiom,
    ! [A2: code_integer] : set_or189985376899183464nteger(A2,A2) = insert_Code_integer(A2,bot_bo3990330152332043303nteger) ).

% atLeastAtMost_singleton
tff(fact_3974_atLeastAtMost__singleton,axiom,
    ! [A2: real] : set_or1222579329274155063t_real(A2,A2) = insert_real(A2,bot_bot_set_real) ).

% atLeastAtMost_singleton
tff(fact_3975_atLeastAtMost__singleton__iff,axiom,
    ! [A2: $o,B2: $o,C2: $o] :
      ( ( set_or8904488021354931149Most_o((A2),(B2)) = insert_o((C2),bot_bot_set_o) )
    <=> ( ( (A2)
        <=> (B2) )
        & ( (B2)
        <=> (C2) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3976_atLeastAtMost__singleton__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A2),B2) = insert_nat(C2,bot_bot_set_nat) )
    <=> ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3977_atLeastAtMost__singleton__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( set_or1266510415728281911st_int(A2,B2) = insert_int(C2,bot_bot_set_int) )
    <=> ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3978_atLeastAtMost__singleton__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( set_or189985376899183464nteger(A2,B2) = insert_Code_integer(C2,bot_bo3990330152332043303nteger) )
    <=> ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3979_atLeastAtMost__singleton__iff,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( set_or1222579329274155063t_real(A2,B2) = insert_real(C2,bot_bot_set_real) )
    <=> ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_3980_atLeastAtMost__singleton_H,axiom,
    ! [A2: $o,B2: $o] :
      ( ( (A2)
      <=> (B2) )
     => ( set_or8904488021354931149Most_o((A2),(B2)) = insert_o((A2),bot_bot_set_o) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3981_atLeastAtMost__singleton_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A2),B2) = insert_nat(A2,bot_bot_set_nat) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3982_atLeastAtMost__singleton_H,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
     => ( set_or1266510415728281911st_int(A2,B2) = insert_int(A2,bot_bot_set_int) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3983_atLeastAtMost__singleton_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 = B2 )
     => ( set_or189985376899183464nteger(A2,B2) = insert_Code_integer(A2,bot_bo3990330152332043303nteger) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3984_atLeastAtMost__singleton_H,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = B2 )
     => ( set_or1222579329274155063t_real(A2,B2) = insert_real(A2,bot_bot_set_real) ) ) ).

% atLeastAtMost_singleton'
tff(fact_3985_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M8: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),Nb)
         => aa_nat_o(P,M8) )
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X2) ) ) ).

% all_nat_less
tff(fact_3986_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M8: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),Nb)
          & aa_nat_o(P,M8) )
    <=> ? [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X2) ) ) ).

% ex_nat_less
tff(fact_3987_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(L),aa_nat_nat(suc,U)) = aa_nat_set_nat(set_or1269000886237332187st_nat(L),U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_3988_atLeastatMost__psubset__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat,D2: set_nat] :
      ( ord_less_set_set_nat(set_or4548717258645045905et_nat(A2,B2),set_or4548717258645045905et_nat(C2,D2))
    <=> ( ( ~ aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
          | ( aa_set_nat_o(ord_less_eq_set_nat(C2),A2)
            & aa_set_nat_o(ord_less_eq_set_nat(B2),D2)
            & ( aa_set_nat_o(ord_less_set_nat(C2),A2)
              | aa_set_nat_o(ord_less_set_nat(B2),D2) ) ) )
        & aa_set_nat_o(ord_less_eq_set_nat(C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3989_atLeastatMost__psubset__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_set_rat_o(ord_less_set_rat(set_or633870826150836451st_rat(A2,B2)),set_or633870826150836451st_rat(C2,D2))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
          | ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),D2)
            & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A2)
              | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),D2) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3990_atLeastatMost__psubset__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( aa_set_num_o(ord_less_set_num(set_or7049704709247886629st_num(A2,B2)),set_or7049704709247886629st_num(C2,D2))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
          | ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A2)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),D2)
            & ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A2)
              | aa_num_o(aa_num_fun_num_o(ord_less_num,B2),D2) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3991_atLeastatMost__psubset__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_set_nat_o(ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A2),B2)),aa_nat_set_nat(set_or1269000886237332187st_nat(C2),D2))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),D2)
            & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A2)
              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),D2) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3992_atLeastatMost__psubset__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_set_int_o(ord_less_set_int(set_or1266510415728281911st_int(A2,B2)),set_or1266510415728281911st_int(C2,D2))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
          | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A2)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),D2)
            & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A2)
              | aa_int_o(aa_int_fun_int_o(ord_less_int,B2),D2) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3993_atLeastatMost__psubset__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(set_or189985376899183464nteger(A2,B2)),set_or189985376899183464nteger(C2,D2))
    <=> ( ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
          | ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A2)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),D2)
            & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,C2),A2)
              | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),D2) ) ) )
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3994_atLeastatMost__psubset__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_set_real_o(ord_less_set_real(set_or1222579329274155063t_real(A2,B2)),set_or1222579329274155063t_real(C2,D2))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
          | ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),A2)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),D2)
            & ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A2)
              | aa_real_o(aa_real_fun_real_o(ord_less_real,B2),D2) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),D2) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_3995_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_3996_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb) = insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_3997_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_nat_nat(suc,Nb))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(M),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_3998_atLeastAtMost__insertL,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( insert_nat(M,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,M)),Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_3999_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat,D2: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A2,B2),set_or3540276404033026485et_nat(C2,D2))
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
       => ( aa_set_nat_o(ord_less_eq_set_nat(C2),A2)
          & aa_set_nat_o(ord_less_set_nat(B2),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4000_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or633870826150836451st_rat(A2,B2)),set_or4029947393144176647an_rat(C2,D2))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4001_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( aa_set_num_o(ord_less_eq_set_num(set_or7049704709247886629st_num(A2,B2)),set_or1222409239386451017an_num(C2,D2))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
       => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A2)
          & aa_num_o(aa_num_fun_num_o(ord_less_num,B2),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4002_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or1222579329274155063t_real(A2,B2)),set_or66887138388493659n_real(C2,D2))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,B2),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4003_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A2),B2)),aa_nat_set_nat(set_or4665077453230672383an_nat(C2),D2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A2)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4004_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(set_or1266510415728281911st_int(A2,B2)),set_or4662586982721622107an_int(C2,D2))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4005_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_or189985376899183464nteger(A2,B2)),set_or8404916559141939852nteger(C2,D2))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,C2),A2)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),D2) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4006_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_set_rat_o(ord_less_eq_set_rat(set_or4029947393144176647an_rat(A2,B2)),set_or633870826150836451st_rat(C2,D2))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),D2) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4007_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(set_or66887138388493659n_real(A2,B2)),set_or1222579329274155063t_real(C2,D2))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),A2)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),D2) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4008_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A2: $o,B2: $o] : set_or7139685690850216873Than_o((A2),(B2)) = minus_minus_set_o(set_or8904488021354931149Most_o((A2),(B2)),insert_o((B2),bot_bot_set_o)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_4009_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A2: real,B2: real] : set_or66887138388493659n_real(A2,B2) = minus_minus_set_real(set_or1222579329274155063t_real(A2,B2),insert_real(B2,bot_bot_set_real)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_4010_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A2: nat,B2: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(A2),B2) = minus_minus_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A2),B2),insert_nat(B2,bot_bot_set_nat)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_4011_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A2: int,B2: int] : set_or4662586982721622107an_int(A2,B2) = minus_minus_set_int(set_or1266510415728281911st_int(A2,B2),insert_int(B2,bot_bot_set_int)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_4012_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A2: code_integer,B2: code_integer] : set_or8404916559141939852nteger(A2,B2) = minus_2355218937544613996nteger(set_or189985376899183464nteger(A2,B2),insert_Code_integer(B2,bot_bo3990330152332043303nteger)) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_4013_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xc: produc7004362771917097900on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_nat,Uv: option_nat] : Xc != produc6699827698519632094on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv))
     => ( ! [Uw2: fun_nat_fun_nat_nat,V2: nat] : Xc != produc6699827698519632094on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F3: fun_nat_fun_nat_nat,A3: nat,B3: nat] : Xc != produc6699827698519632094on_nat(F3,produc5098337634421038937on_nat(some_nat(A3),some_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_4014_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [Xc: produc1260151622200656695at_nat] :
      ( ! [Uu2: fun_Pr2477239776880934807at_nat,Uv: option4927543243414619207at_nat] : Xc != produc7814092408141220007at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
     => ( ! [Uw2: fun_Pr2477239776880934807at_nat,V2: product_prod_nat_nat] : Xc != produc7814092408141220007at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F3: fun_Pr2477239776880934807at_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] : Xc != produc7814092408141220007at_nat(F3,produc488173922507101015at_nat(some_P7363390416028606310at_nat(A3),some_P7363390416028606310at_nat(B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_4015_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xc: produc1615844007061411358on_nat] :
      ( ! [Uu2: fun_nat_fun_nat_o,Uv: option_nat] : Xc != produc7967433646104525454on_nat(Uu2,produc5098337634421038937on_nat(none_nat,Uv))
     => ( ! [Uw2: fun_nat_fun_nat_o,V2: nat] : Xc != produc7967433646104525454on_nat(Uw2,produc5098337634421038937on_nat(some_nat(V2),none_nat))
       => ~ ! [F3: fun_nat_fun_nat_o,X3: nat,Y: nat] : Xc != produc7967433646104525454on_nat(F3,produc5098337634421038937on_nat(some_nat(X3),some_nat(Y))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_4016_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [Xc: produc3739950874144486240at_nat] :
      ( ! [Uu2: fun_Pr2028404973800837934_nat_o,Uv: option4927543243414619207at_nat] : Xc != produc3653971535690468304at_nat(Uu2,produc488173922507101015at_nat(none_P5556105721700978146at_nat,Uv))
     => ( ! [Uw2: fun_Pr2028404973800837934_nat_o,V2: product_prod_nat_nat] : Xc != produc3653971535690468304at_nat(Uw2,produc488173922507101015at_nat(some_P7363390416028606310at_nat(V2),none_P5556105721700978146at_nat))
       => ~ ! [F3: fun_Pr2028404973800837934_nat_o,X3: product_prod_nat_nat,Y: product_prod_nat_nat] : Xc != produc3653971535690468304at_nat(F3,produc488173922507101015at_nat(some_P7363390416028606310at_nat(X3),some_P7363390416028606310at_nat(Y))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_4017_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F2: fun_Pr2477239776880934807at_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(F2,some_P7363390416028606310at_nat(A2),some_P7363390416028606310at_nat(B2)) = some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(F2,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_4018_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F2: fun_nat_fun_nat_nat,A2: nat,B2: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(F2),some_nat(A2)),some_nat(B2)) = some_nat(aa_nat_nat(aa_nat_fun_nat_nat(F2,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_4019_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_Pr2477239776880934807at_nat,Uv2: option4927543243414619207at_nat] : vEBT_V1502963449132264192at_nat(Uu,none_P5556105721700978146at_nat,Uv2) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_4020_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: fun_nat_fun_nat_nat,Uv2: option_nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uu),none_nat),Uv2) = none_nat ).

% VEBT_internal.option_shift.simps(1)
tff(fact_4021_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xc: fun_Pr2477239776880934807at_nat,Xaa: option4927543243414619207at_nat,Xba: option4927543243414619207at_nat,Ya: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat(Xc,Xaa,Xba) = Ya )
     => ( ( ( Xaa = none_P5556105721700978146at_nat )
         => ( Ya != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] : Xaa = some_P7363390416028606310at_nat(V2)
           => ( ( Xba = none_P5556105721700978146at_nat )
             => ( Ya != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xaa = some_P7363390416028606310at_nat(A3) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xba = some_P7363390416028606310at_nat(B3) )
                   => ( Ya != some_P7363390416028606310at_nat(aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(Xc,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_4022_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [Xc: fun_nat_fun_nat_nat,Xaa: option_nat,Xba: option_nat,Ya: option_nat] :
      ( ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Xc),Xaa),Xba) = Ya )
     => ( ( ( Xaa = none_nat )
         => ( Ya != none_nat ) )
       => ( ( ? [V2: nat] : Xaa = some_nat(V2)
           => ( ( Xba = none_nat )
             => ( Ya != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xaa = some_nat(A3) )
               => ! [B3: nat] :
                    ( ( Xba = some_nat(B3) )
                   => ( Ya != some_nat(aa_nat_nat(aa_nat_fun_nat_nat(Xc,A3),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_4023_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_Pr2477239776880934807at_nat,V: product_prod_nat_nat] : vEBT_V1502963449132264192at_nat(Uw,some_P7363390416028606310at_nat(V),none_P5556105721700978146at_nat) = none_P5556105721700978146at_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_4024_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: fun_nat_fun_nat_nat,V: nat] : aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_V4262088993061758097ft_nat(Uw),some_nat(V)),none_nat) = none_nat ).

% VEBT_internal.option_shift.simps(2)
tff(fact_4025_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw)) = none_nat ).

% vebt_mint.simps(2)
tff(fact_4026_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw)) = none_nat ).

% vebt_maxt.simps(2)
tff(fact_4027_vebt__mint_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : vEBT_vebt_mint(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Ux2,Uy2),Uz2)) = some_nat(Mia) ).

% vebt_mint.simps(3)
tff(fact_4028_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : vEBT_vebt_maxt(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Ux2,Uy2),Uz2)) = some_nat(Maa) ).

% vebt_maxt.simps(3)
tff(fact_4029_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_mint(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))) = $ite(
        (A2),
        some_nat(zero_zero_nat),
        $ite((B2),some_nat(one_one_nat),none_nat) ) ).

% vebt_mint.simps(1)
tff(fact_4030_VEBT__internal_OTb_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,zero_zero_nat)) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_4031_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_4032_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_4033_vebt__delete_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,Xc: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,zero_zero_nat),Tr),Sm) ).

% vebt_delete.simps(6)
tff(fact_4034_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_4035_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xc: nat,Ya: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xc),Ya)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc)
         => aa_int_o(P,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,Xc)),aa_nat_int(semiri1314217659103216013at_int,Ya))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_4036_heaphelp,axiom,
    ! [Xaa: array_VEBT_VEBTi,Tree_isa: list_VEBT_VEBTi,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xba: vEBT_VEBTi,Nb: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Xaa),Tree_isa)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_isa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),Xba))),
                  aa_o_assn(pure_assn,
                    ( ( none_nat = none_nat )
                    & ( Nb = Nb ) )))),aa_o_assn(pure_assn,Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Nb,Xaa),Xba)))),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Nb,TreeLista),Summarya)),Xc)),H) ) ).

% heaphelp
tff(fact_4037_heaphelp,axiom,
    ! [Xaa: array_VEBT_VEBTi,Tree_isa: list_VEBT_VEBTi,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xba: vEBT_VEBTi,Nb: nat,Xc: vEBT_VEBTi,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Xaa),Tree_isa)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,TreeLista),Tree_isa))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),Xba))),
                  aa_o_assn(pure_assn,
                    ( ( none_P5556105721700978146at_nat = none_P5556105721700978146at_nat )
                    & ( Nb = Nb ) )))),aa_o_assn(pure_assn,Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Nb,Xaa),Xba)))),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Nb,TreeLista),Summarya)),Xc)),H) ) ).

% heaphelp
tff(fact_4038_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N: nat] :
              ( aa_nat_o(P,N)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct2
tff(fact_4039_VEBTi_Oinject_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12: nat,X13a: array_VEBT_VEBTi,X14a: vEBT_VEBTi,Y11: option4927543243414619207at_nat,Y12: nat,Y13: array_VEBT_VEBTi,Y14: vEBT_VEBTi] :
      ( ( aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11a,X12,X13a),X14a) = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Y11,Y12,Y13),Y14) )
    <=> ( ( X11a = Y11 )
        & ( X12 = Y12 )
        & ( X13a = Y13 )
        & ( X14a = Y14 ) ) ) ).

% VEBTi.inject(1)
tff(fact_4040_Rep__assn__inject,axiom,
    ! [Xc: assn,Ya: assn] :
      ( ( rep_assn(Xc) = rep_assn(Ya) )
    <=> ( Xc = Ya ) ) ).

% Rep_assn_inject
tff(fact_4041_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(P),Q)),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(rep_assn(Q),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(5)
tff(fact_4042_mod__pure__star__dist,axiom,
    ! [P: assn,B2: $o,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2)))),H)
    <=> ( aa_Pro653082453696054616_nat_o(rep_assn(P),H)
        & (B2) ) ) ).

% mod_pure_star_dist
tff(fact_4043_mod__h__bot__iff_I1_J,axiom,
    ! [B2: $o,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_o_assn(pure_assn,(B2))),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> (B2) ) ).

% mod_h_bot_iff(1)
tff(fact_4044_mod__h__bot__iff_I4_J,axiom,
    ! [Q3: array_VEBT_VEBTi,Ya: list_VEBT_VEBTi,H: heap_e7401611519738050253t_unit] : ~ aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Q3),Ya)),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ).

% mod_h_bot_iff(4)
tff(fact_4045_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B2: $o] :
      ( entails(P,aa_assn_assn(times_times_assn(Q),aa_o_assn(pure_assn,(B2))))
    <=> ( ! [H3: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),H3)
           => (B2) )
        & entails(P,Q) ) ) ).

% ent_pure_post_iff
tff(fact_4046_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B2: $o] :
      ( entails(P,aa_o_assn(pure_assn,(B2)))
    <=> ( ! [H3: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),H3)
           => (B2) )
        & entails(P,one_one_assn) ) ) ).

% ent_pure_post_iff_sng
tff(fact_4047_mod__starE,axiom,
    ! [A2: assn,B2: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(A2),B2)),H)
     => ~ ( ? [X_12: produc3658429121746597890et_nat] : aa_Pro653082453696054616_nat_o(rep_assn(A2),X_12)
         => ! [H_2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(B2),H_2) ) ) ).

% mod_starE
tff(fact_4048_mod__starD,axiom,
    ! [A: assn,B: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(A),B)),H)
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(A),H1)
          & aa_Pro653082453696054616_nat_o(rep_assn(B),H22) ) ) ).

% mod_starD
tff(fact_4049_entails__def,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
    <=> ! [H3: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H3)
         => aa_Pro653082453696054616_nat_o(rep_assn(Q),H3) ) ) ).

% entails_def
tff(fact_4050_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(P),H4)
         => aa_Pro653082453696054616_nat_o(rep_assn(Q),H4) )
     => entails(P,Q) ) ).

% entailsI
tff(fact_4051_entailsD,axiom,
    ! [P: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( entails(P,Q)
     => ( aa_Pro653082453696054616_nat_o(rep_assn(P),H)
       => aa_Pro653082453696054616_nat_o(rep_assn(Q),H) ) ) ).

% entailsD
tff(fact_4052_ent__fwd,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat,Q: assn] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P),H)
     => ( entails(P,Q)
       => aa_Pro653082453696054616_nat_o(rep_assn(Q),H) ) ) ).

% ent_fwd
tff(fact_4053_mod__h__bot__indep,axiom,
    ! [P: assn,H: heap_e7401611519738050253t_unit,H2: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ) ).

% mod_h_bot_indep
tff(fact_4054_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(plus_plus_int(U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_4055_simp__from__to,axiom,
    ! [I: int,J2: int] :
      set_or1266510415728281911st_int(I,J2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J2),I),bot_bot_set_int,insert_int(I,set_or1266510415728281911st_int(aa_int_int(plus_plus_int(I),one_one_int),J2))) ).

% simp_from_to
tff(fact_4056_aset_I2_J,axiom,
    ! [D: int,A: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),A)
                 => ( X3 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(plus_plus_int(X3),D)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),A)
                   => ( X3 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(plus_plus_int(X3),D)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(plus_plus_int(X4),D))
                | aa_int_o(Q,aa_int_int(plus_plus_int(X4),D)) ) ) ) ) ) ).

% aset(2)
tff(fact_4057_aset_I1_J,axiom,
    ! [D: int,A: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),A)
                 => ( X3 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(plus_plus_int(X3),D)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),A)
                   => ( X3 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(plus_plus_int(X3),D)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(plus_plus_int(X4),D))
                & aa_int_o(Q,aa_int_int(plus_plus_int(X4),D)) ) ) ) ) ) ).

% aset(1)
tff(fact_4058_bset_I2_J,axiom,
    ! [D: int,B: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),B)
                 => ( X3 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(minus_minus_int(X3),D)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),B)
                   => ( X3 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(minus_minus_int(X3),D)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(minus_minus_int(X4),D))
                | aa_int_o(Q,aa_int_int(minus_minus_int(X4),D)) ) ) ) ) ) ).

% bset(2)
tff(fact_4059_bset_I1_J,axiom,
    ! [D: int,B: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb: int] :
                  ( aa_set_int_o(member_int(Xb),B)
                 => ( X3 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(minus_minus_int(X3),D)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb: int] :
                    ( aa_set_int_o(member_int(Xb),B)
                   => ( X3 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(minus_minus_int(X3),D)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(minus_minus_int(X4),D))
                & aa_int_o(Q,aa_int_int(minus_minus_int(X4),D)) ) ) ) ) ) ).

% bset(1)
tff(fact_4060_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L1279224858307276611T_VEBT(A,Xs,Xsi)),H)
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_s6755466524823107622T_VEBT(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_4061_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_re8050108611700907676T_assn,Xs: list_real,Xsi: list_VEBT_VEBT,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L4595930785310033027T_VEBT(A,Xs,Xsi)),H)
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_4062_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_o_4353168661238401050T_assn,Xs: list_o,Xsi: list_VEBT_VEBT,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L1750719106661372127T_VEBT(A,Xs,Xsi)),H)
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_size_list_o(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_4063_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_na4489259242115493880T_assn,Xs: list_nat,Xsi: list_VEBT_VEBT,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L8158188754432654943T_VEBT(A,Xs,Xsi)),H)
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_size_list_nat(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_4064_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_VE5577048379444288083T_assn,Xs: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L7265847600308530106T_VEBT(A,Xs,Xsi)),H)
     => ( size_s6755466524823107622T_VEBT(Xsi) = size_s7982070591426661849_VEBTi(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_4065_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_VE6910866152645720856l_assn,Xs: list_VEBT_VEBT,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L5781919052683127133T_real(A,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_s6755466524823107622T_VEBT(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_4066_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_re6303261851654292474l_assn,Xs: list_real,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L1930518968523514909l_real(A,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_size_list_real(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_4067_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_o_fun_real_assn,Xs: list_o,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L4725278957065240257o_real(A,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_size_list_o(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_4068_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_na448437756743965086l_assn,Xs: list_nat,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L6102073776069194049t_real(A,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_size_list_nat(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_4069_extract__pre__list__assn__lengthD,axiom,
    ! [A: fun_VE1885522831824361987l_assn,Xs: list_VEBT_VEBTi,Xsi: list_real,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(vEBT_L8937798142398754470i_real(A,Xs,Xsi)),H)
     => ( size_size_list_real(Xsi) = size_s7982070591426661849_VEBTi(Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_4070_mod__emp__simp,axiom,
    ! [H: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(one_one_assn),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ).

% mod_emp_simp
tff(fact_4071_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),aa_int_int(plus_plus_int(one_one_int),Nb))
     => ( set_or1266510415728281911st_int(M,aa_int_int(plus_plus_int(one_one_int),Nb)) = insert_int(aa_int_int(plus_plus_int(one_one_int),Nb),set_or1266510415728281911st_int(M,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_4072_pinf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P2,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q2,X3) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
           => ( ( aa_real_o(P,X4)
                & aa_real_o(Q,X4) )
            <=> ( aa_real_o(P2,X4)
                & aa_real_o(Q2,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_4073_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P2,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
           => ( ( aa_rat_o(P,X4)
                & aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P2,X4)
                & aa_rat_o(Q2,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_4074_pinf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P2,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X4)
           => ( ( aa_num_o(P,X4)
                & aa_num_o(Q,X4) )
            <=> ( aa_num_o(P2,X4)
                & aa_num_o(Q2,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_4075_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P2,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
           => ( ( aa_nat_o(P,X4)
                & aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P2,X4)
                & aa_nat_o(Q2,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_4076_pinf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P2,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
            <=> ( aa_int_o(P2,X4)
                & aa_int_o(Q2,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_4077_pinf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P2,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z3),X3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q2,X3) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
           => ( ( aa_real_o(P,X4)
                | aa_real_o(Q,X4) )
            <=> ( aa_real_o(P2,X4)
                | aa_real_o(Q2,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_4078_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P2,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z3),X3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
           => ( ( aa_rat_o(P,X4)
                | aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P2,X4)
                | aa_rat_o(Q2,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_4079_pinf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P2,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z3),X3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X4)
           => ( ( aa_num_o(P,X4)
                | aa_num_o(Q,X4) )
            <=> ( aa_num_o(P2,X4)
                | aa_num_o(Q2,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_4080_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P2,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z3),X3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
           => ( ( aa_nat_o(P,X4)
                | aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P2,X4)
                | aa_nat_o(Q2,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_4081_pinf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P2,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
            <=> ( aa_int_o(P2,X4)
                | aa_int_o(Q2,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_4082_pinf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => ( X4 != Ta ) ) ).

% pinf(3)
tff(fact_4083_pinf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ( X4 != Ta ) ) ).

% pinf(3)
tff(fact_4084_pinf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X4)
     => ( X4 != Ta ) ) ).

% pinf(3)
tff(fact_4085_pinf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => ( X4 != Ta ) ) ).

% pinf(3)
tff(fact_4086_pinf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ( X4 != Ta ) ) ).

% pinf(3)
tff(fact_4087_pinf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => ( X4 != Ta ) ) ).

% pinf(4)
tff(fact_4088_pinf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ( X4 != Ta ) ) ).

% pinf(4)
tff(fact_4089_pinf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X4)
     => ( X4 != Ta ) ) ).

% pinf(4)
tff(fact_4090_pinf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => ( X4 != Ta ) ) ).

% pinf(4)
tff(fact_4091_pinf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ( X4 != Ta ) ) ).

% pinf(4)
tff(fact_4092_pinf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Ta) ) ).

% pinf(5)
tff(fact_4093_pinf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Ta) ) ).

% pinf(5)
tff(fact_4094_pinf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X4)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Ta) ) ).

% pinf(5)
tff(fact_4095_pinf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Ta) ) ).

% pinf(5)
tff(fact_4096_pinf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Ta) ) ).

% pinf(5)
tff(fact_4097_pinf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X4) ) ).

% pinf(7)
tff(fact_4098_pinf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X4) ) ).

% pinf(7)
tff(fact_4099_pinf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X4)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X4) ) ).

% pinf(7)
tff(fact_4100_pinf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X4) ) ).

% pinf(7)
tff(fact_4101_pinf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X4) ) ).

% pinf(7)
tff(fact_4102_minf_I1_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P2,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q2,X3) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
           => ( ( aa_real_o(P,X4)
                & aa_real_o(Q,X4) )
            <=> ( aa_real_o(P2,X4)
                & aa_real_o(Q2,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_4103_minf_I1_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P2,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
           => ( ( aa_rat_o(P,X4)
                & aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P2,X4)
                & aa_rat_o(Q2,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_4104_minf_I1_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P2,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z2)
           => ( ( aa_num_o(P,X4)
                & aa_num_o(Q,X4) )
            <=> ( aa_num_o(P2,X4)
                & aa_num_o(Q2,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_4105_minf_I1_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P2,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
           => ( ( aa_nat_o(P,X4)
                & aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P2,X4)
                & aa_nat_o(Q2,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_4106_minf_I1_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P2,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
            <=> ( aa_int_o(P2,X4)
                & aa_int_o(Q2,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_4107_minf_I2_J,axiom,
    ! [P: fun_real_o,P2: fun_real_o,Q: fun_real_o,Q2: fun_real_o] :
      ( ? [Z3: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P2,X3) ) )
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q2,X3) ) )
       => ? [Z2: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
           => ( ( aa_real_o(P,X4)
                | aa_real_o(Q,X4) )
            <=> ( aa_real_o(P2,X4)
                | aa_real_o(Q2,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_4108_minf_I2_J,axiom,
    ! [P: fun_rat_o,P2: fun_rat_o,Q: fun_rat_o,Q2: fun_rat_o] :
      ( ? [Z3: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P2,X3) ) )
     => ( ? [Z3: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q2,X3) ) )
       => ? [Z2: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
           => ( ( aa_rat_o(P,X4)
                | aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P2,X4)
                | aa_rat_o(Q2,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_4109_minf_I2_J,axiom,
    ! [P: fun_num_o,P2: fun_num_o,Q: fun_num_o,Q2: fun_num_o] :
      ( ? [Z3: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P2,X3) ) )
     => ( ? [Z3: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q2,X3) ) )
       => ? [Z2: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z2)
           => ( ( aa_num_o(P,X4)
                | aa_num_o(Q,X4) )
            <=> ( aa_num_o(P2,X4)
                | aa_num_o(Q2,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_4110_minf_I2_J,axiom,
    ! [P: fun_nat_o,P2: fun_nat_o,Q: fun_nat_o,Q2: fun_nat_o] :
      ( ? [Z3: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P2,X3) ) )
     => ( ? [Z3: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q2,X3) ) )
       => ? [Z2: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
           => ( ( aa_nat_o(P,X4)
                | aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P2,X4)
                | aa_nat_o(Q2,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_4111_minf_I2_J,axiom,
    ! [P: fun_int_o,P2: fun_int_o,Q: fun_int_o,Q2: fun_int_o] :
      ( ? [Z3: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P2,X3) ) )
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q2,X3) ) )
       => ? [Z2: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
            <=> ( aa_int_o(P2,X4)
                | aa_int_o(Q2,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_4112_minf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ( X4 != Ta ) ) ).

% minf(3)
tff(fact_4113_minf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ( X4 != Ta ) ) ).

% minf(3)
tff(fact_4114_minf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z2)
     => ( X4 != Ta ) ) ).

% minf(3)
tff(fact_4115_minf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ( X4 != Ta ) ) ).

% minf(3)
tff(fact_4116_minf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ( X4 != Ta ) ) ).

% minf(3)
tff(fact_4117_minf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ( X4 != Ta ) ) ).

% minf(4)
tff(fact_4118_minf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ( X4 != Ta ) ) ).

% minf(4)
tff(fact_4119_minf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z2)
     => ( X4 != Ta ) ) ).

% minf(4)
tff(fact_4120_minf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ( X4 != Ta ) ) ).

% minf(4)
tff(fact_4121_minf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ( X4 != Ta ) ) ).

% minf(4)
tff(fact_4122_minf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Ta) ) ).

% minf(5)
tff(fact_4123_minf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Ta) ) ).

% minf(5)
tff(fact_4124_minf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Ta) ) ).

% minf(5)
tff(fact_4125_minf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Ta) ) ).

% minf(5)
tff(fact_4126_minf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Ta) ) ).

% minf(5)
tff(fact_4127_minf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X4) ) ).

% minf(7)
tff(fact_4128_minf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X4) ) ).

% minf(7)
tff(fact_4129_minf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X4) ) ).

% minf(7)
tff(fact_4130_minf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X4) ) ).

% minf(7)
tff(fact_4131_minf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X4) ) ).

% minf(7)
tff(fact_4132_bset_I3_J,axiom,
    ! [D: int,Ta: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( aa_set_int_o(member_int(aa_int_int(minus_minus_int(Ta),one_one_int)),B)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( X4 = Ta )
             => ( aa_int_int(minus_minus_int(X4),D) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_4133_bset_I4_J,axiom,
    ! [D: int,Ta: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( aa_set_int_o(member_int(Ta),B)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( ( X4 != Ta )
             => ( aa_int_int(minus_minus_int(X4),D) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_4134_bset_I5_J,axiom,
    ! [D: int,B: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B)
                 => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(X4),D)),Ta) ) ) ) ).

% bset(5)
tff(fact_4135_bset_I7_J,axiom,
    ! [D: int,Ta: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( aa_set_int_o(member_int(Ta),B)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(minus_minus_int(X4),D)) ) ) ) ) ).

% bset(7)
tff(fact_4136_aset_I3_J,axiom,
    ! [D: int,Ta: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( aa_set_int_o(member_int(aa_int_int(plus_plus_int(Ta),one_one_int)),A)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( X4 = Ta )
             => ( aa_int_int(plus_plus_int(X4),D) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_4137_aset_I4_J,axiom,
    ! [D: int,Ta: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( aa_set_int_o(member_int(Ta),A)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( ( X4 != Ta )
             => ( aa_int_int(plus_plus_int(X4),D) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_4138_aset_I5_J,axiom,
    ! [D: int,Ta: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( aa_set_int_o(member_int(Ta),A)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(X4),D)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_4139_aset_I7_J,axiom,
    ! [D: int,A: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A)
                 => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X4)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(plus_plus_int(X4),D)) ) ) ) ).

% aset(7)
tff(fact_4140_periodic__finite__ex,axiom,
    ! [D2: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P,X3)
          <=> aa_int_o(P,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D2))) )
       => ( ? [X_1: int] : aa_int_o(P,X_1)
        <=> ? [X2: int] :
              ( aa_set_int_o(member_int(X2),set_or1266510415728281911st_int(one_one_int,D2))
              & aa_int_o(P,X2) ) ) ) ) ).

% periodic_finite_ex
tff(fact_4141_bset_I6_J,axiom,
    ! [D: int,B: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B)
                 => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(X4),D)),Ta) ) ) ) ).

% bset(6)
tff(fact_4142_bset_I8_J,axiom,
    ! [D: int,Ta: int,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( aa_set_int_o(member_int(aa_int_int(minus_minus_int(Ta),one_one_int)),B)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),B)
                   => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(minus_minus_int(X4),D)) ) ) ) ) ).

% bset(8)
tff(fact_4143_aset_I6_J,axiom,
    ! [D: int,Ta: int,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( aa_set_int_o(member_int(aa_int_int(plus_plus_int(Ta),one_one_int)),A)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
               => ! [Xb2: int] :
                    ( aa_set_int_o(member_int(Xb2),A)
                   => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(X4),D)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_4144_aset_I8_J,axiom,
    ! [D: int,A: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A)
                 => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(plus_plus_int(X4),D)) ) ) ) ).

% aset(8)
tff(fact_4145_cppi,axiom,
    ! [D: int,P: fun_int_o,P2: fun_int_o,A: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P2,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D))
                 => ! [Xb: int] :
                      ( aa_set_int_o(member_int(Xb),A)
                     => ( X3 != aa_int_int(minus_minus_int(Xb),Xa) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(plus_plus_int(X3),D)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P2,X3)
              <=> aa_int_o(P2,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D))) )
           => ( ? [X_1: int] : aa_int_o(P,X_1)
            <=> ( ? [X2: int] :
                    ( aa_set_int_o(member_int(X2),set_or1266510415728281911st_int(one_one_int,D))
                    & aa_int_o(P2,X2) )
                | ? [X2: int] :
                    ( aa_set_int_o(member_int(X2),set_or1266510415728281911st_int(one_one_int,D))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),A)
                        & aa_int_o(P,aa_int_int(minus_minus_int(Xa3),X2)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4146_cpmi,axiom,
    ! [D: int,P: fun_int_o,P2: fun_int_o,B: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P2,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),set_or1266510415728281911st_int(one_one_int,D))
                 => ! [Xb: int] :
                      ( aa_set_int_o(member_int(Xb),B)
                     => ( X3 != aa_int_int(plus_plus_int(Xb),Xa) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(minus_minus_int(X3),D)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P2,X3)
              <=> aa_int_o(P2,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D))) )
           => ( ? [X_1: int] : aa_int_o(P,X_1)
            <=> ( ? [X2: int] :
                    ( aa_set_int_o(member_int(X2),set_or1266510415728281911st_int(one_one_int,D))
                    & aa_int_o(P2,X2) )
                | ? [X2: int] :
                    ( aa_set_int_o(member_int(X2),set_or1266510415728281911st_int(one_one_int,D))
                    & ? [Xa3: int] :
                        ( aa_set_int_o(member_int(Xa3),B)
                        & aa_int_o(P,aa_int_int(plus_plus_int(Xa3),X2)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4147_pinf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Ta) ) ).

% pinf(6)
tff(fact_4148_pinf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Ta) ) ).

% pinf(6)
tff(fact_4149_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X4)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Ta) ) ).

% pinf(6)
tff(fact_4150_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ta) ) ).

% pinf(6)
tff(fact_4151_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Ta) ) ).

% pinf(6)
tff(fact_4152_pinf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X4) ) ).

% pinf(8)
tff(fact_4153_pinf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X4) ) ).

% pinf(8)
tff(fact_4154_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X4)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X4) ) ).

% pinf(8)
tff(fact_4155_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X4) ) ).

% pinf(8)
tff(fact_4156_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X4) ) ).

% pinf(8)
tff(fact_4157_minf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Ta) ) ).

% minf(6)
tff(fact_4158_minf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Ta) ) ).

% minf(6)
tff(fact_4159_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Ta) ) ).

% minf(6)
tff(fact_4160_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ta) ) ).

% minf(6)
tff(fact_4161_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Ta) ) ).

% minf(6)
tff(fact_4162_minf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X4) ) ).

% minf(8)
tff(fact_4163_minf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X4) ) ).

% minf(8)
tff(fact_4164_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z2: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X4) ) ).

% minf(8)
tff(fact_4165_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X4) ) ).

% minf(8)
tff(fact_4166_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X4) ) ).

% minf(8)
tff(fact_4167_inf__period_I1_J,axiom,
    ! [P: fun_real_o,D: real,Q: fun_real_o] :
      ( ! [X3: real,K2: real] :
          ( aa_real_o(P,X3)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D))) )
     => ( ! [X3: real,K2: real] :
            ( aa_real_o(Q,X3)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D))) )
       => ! [X4: real,K5: real] :
            ( ( aa_real_o(P,X4)
              & aa_real_o(Q,X4) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D)))
              & aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D))) ) ) ) ) ).

% inf_period(1)
tff(fact_4168_inf__period_I1_J,axiom,
    ! [P: fun_rat_o,D: rat,Q: fun_rat_o] :
      ( ! [X3: rat,K2: rat] :
          ( aa_rat_o(P,X3)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D))) )
     => ( ! [X3: rat,K2: rat] :
            ( aa_rat_o(Q,X3)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D))) )
       => ! [X4: rat,K5: rat] :
            ( ( aa_rat_o(P,X4)
              & aa_rat_o(Q,X4) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K5),D)))
              & aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K5),D))) ) ) ) ) ).

% inf_period(1)
tff(fact_4169_inf__period_I1_J,axiom,
    ! [P: fun_int_o,D: int,Q: fun_int_o] :
      ( ! [X3: int,K2: int] :
          ( aa_int_o(P,X3)
        <=> aa_int_o(P,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D))) )
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(Q,X3)
          <=> aa_int_o(Q,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D))) )
       => ! [X4: int,K5: int] :
            ( ( aa_int_o(P,X4)
              & aa_int_o(Q,X4) )
          <=> ( aa_int_o(P,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D)))
              & aa_int_o(Q,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D))) ) ) ) ) ).

% inf_period(1)
tff(fact_4170_inf__period_I2_J,axiom,
    ! [P: fun_real_o,D: real,Q: fun_real_o] :
      ( ! [X3: real,K2: real] :
          ( aa_real_o(P,X3)
        <=> aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D))) )
     => ( ! [X3: real,K2: real] :
            ( aa_real_o(Q,X3)
          <=> aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K2),D))) )
       => ! [X4: real,K5: real] :
            ( ( aa_real_o(P,X4)
              | aa_real_o(Q,X4) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D)))
              | aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D))) ) ) ) ) ).

% inf_period(2)
tff(fact_4171_inf__period_I2_J,axiom,
    ! [P: fun_rat_o,D: rat,Q: fun_rat_o] :
      ( ! [X3: rat,K2: rat] :
          ( aa_rat_o(P,X3)
        <=> aa_rat_o(P,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D))) )
     => ( ! [X3: rat,K2: rat] :
            ( aa_rat_o(Q,X3)
          <=> aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X3),aa_rat_rat(times_times_rat(K2),D))) )
       => ! [X4: rat,K5: rat] :
            ( ( aa_rat_o(P,X4)
              | aa_rat_o(Q,X4) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K5),D)))
              | aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K5),D))) ) ) ) ) ).

% inf_period(2)
tff(fact_4172_inf__period_I2_J,axiom,
    ! [P: fun_int_o,D: int,Q: fun_int_o] :
      ( ! [X3: int,K2: int] :
          ( aa_int_o(P,X3)
        <=> aa_int_o(P,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D))) )
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(Q,X3)
          <=> aa_int_o(Q,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D))) )
       => ! [X4: int,K5: int] :
            ( ( aa_int_o(P,X4)
              | aa_int_o(Q,X4) )
          <=> ( aa_int_o(P,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D)))
              | aa_int_o(Q,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D))) ) ) ) ) ).

% inf_period(2)
tff(fact_4173_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [Xc: nat] :
      ( ( Xc != zero_zero_nat )
     => ( ( Xc != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va2: nat] : Xc != aa_nat_nat(suc,aa_nat_nat(suc,Va2)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_4174_vebt__delete_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Nb: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2)) ).

% vebt_delete.simps(3)
tff(fact_4175_vebt__delete_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),zero_zero_nat) = aa_o_VEBT_VEBT(vEBT_Leaf($false),(B2)) ).

% vebt_delete.simps(1)
tff(fact_4176_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,zero_zero_nat) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
tff(fact_4177_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
tff(fact_4178_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Uu: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya)),Uu) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeLista),Summarya) ).

% vebt_delete.simps(4)
tff(fact_4179_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,zero_zero_nat) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
tff(fact_4180_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))) = one_one_real ).

% VEBT_internal.cnt.simps(1)
tff(fact_4181_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),aa_nat_nat(suc,zero_zero_nat)) = aa_o_VEBT_VEBT(vEBT_Leaf((A2)),$false) ).

% vebt_delete.simps(2)
tff(fact_4182_minusinfinity,axiom,
    ! [D2: int,P1: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P1,X3)
          <=> aa_int_o(P1,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D2))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z3)
             => ( aa_int_o(P,X3)
              <=> aa_int_o(P1,X3) ) )
         => ( ? [X_13: int] : aa_int_o(P1,X_13)
           => ? [X_12: int] : aa_int_o(P,X_12) ) ) ) ) ).

% minusinfinity
tff(fact_4183_plusinfinity,axiom,
    ! [D2: int,P2: fun_int_o,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(P2,X3)
          <=> aa_int_o(P2,aa_int_int(minus_minus_int(X3),aa_int_int(times_times_int(K2),D2))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),X3)
             => ( aa_int_o(P,X3)
              <=> aa_int_o(P2,X3) ) )
         => ( ? [X_13: int] : aa_int_o(P2,X_13)
           => ? [X_12: int] : aa_int_o(P,X_12) ) ) ) ) ).

% plusinfinity
tff(fact_4184_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
tff(fact_4185_double__not__eq__Suc__double,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_4186_Suc__double__not__eq__double,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_4187_incr__mult__lemma,axiom,
    ! [D2: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int] :
            ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(plus_plus_int(X3),D2)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X4: int] :
              ( aa_int_o(P,X4)
             => aa_int_o(P,aa_int_int(plus_plus_int(X4),aa_int_int(times_times_int(K),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_4188_decr__mult__lemma,axiom,
    ! [D2: int,P: fun_int_o,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X3: int] :
            ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(minus_minus_int(X3),D2)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X4: int] :
              ( aa_int_o(P,X4)
             => aa_int_o(P,aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_4189_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8646137997579335489_i_l_d,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_4190_vebt__delete_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,Xc: nat] : aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,TrLst),Smry)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,TrLst),Smry) ).

% vebt_delete.simps(5)
tff(fact_4191_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_V8346862874174094_d_u_p,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_4192_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    aa_nat_nat(vEBT_VEBT_Tb2,zero_zero_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_4193_VEBT__internal_OTb_Osimps_I1_J,axiom,
    aa_nat_int(vEBT_VEBT_Tb,zero_zero_nat) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_4194_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% VEBT_internal.space'.simps(1)
tff(fact_4195_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ).

% VEBT_internal.space.simps(1)
tff(fact_4196_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_4197_nat__approx__posE,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ~ ! [N: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N)))),E) ) ).

% nat_approx_posE
tff(fact_4198_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xc: vEBT_VEBT,Ya: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xc)
      <=> (Ya) )
     => ( ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) )
         => ~ (Ya) )
       => ( ( ? [Uv: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf($true),(Uv))
           => (Ya) )
         => ( ( ? [Uu2: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),$true)
             => (Ya) )
           => ( ( ? [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux),Uy)
               => ~ (Ya) )
             => ~ ( ? [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc)
                 => (Ya) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_4199_member__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_m_e_m_b_e_r(Ta),Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_4200_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xc: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),Xc) ).

% vebt_member.simps(4)
tff(fact_4201_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)),zero_zero_int,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2))) ).

% int_ops(6)
tff(fact_4202_max__bot2,axiom,
    ! [Xc: set_real] : ord_max_set_real(Xc,bot_bot_set_real) = Xc ).

% max_bot2
tff(fact_4203_max__bot2,axiom,
    ! [Xc: set_o] : ord_max_set_o(Xc,bot_bot_set_o) = Xc ).

% max_bot2
tff(fact_4204_max__bot2,axiom,
    ! [Xc: set_nat] : ord_max_set_nat(Xc,bot_bot_set_nat) = Xc ).

% max_bot2
tff(fact_4205_max__bot2,axiom,
    ! [Xc: set_int] : ord_max_set_int(Xc,bot_bot_set_int) = Xc ).

% max_bot2
tff(fact_4206_max__bot2,axiom,
    ! [Xc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),bot_bot_nat) = Xc ).

% max_bot2
tff(fact_4207_max__bot,axiom,
    ! [Xc: set_real] : ord_max_set_real(bot_bot_set_real,Xc) = Xc ).

% max_bot
tff(fact_4208_max__bot,axiom,
    ! [Xc: set_o] : ord_max_set_o(bot_bot_set_o,Xc) = Xc ).

% max_bot
tff(fact_4209_max__bot,axiom,
    ! [Xc: set_nat] : ord_max_set_nat(bot_bot_set_nat,Xc) = Xc ).

% max_bot
tff(fact_4210_max__bot,axiom,
    ! [Xc: set_int] : ord_max_set_int(bot_bot_set_int,Xc) = Xc ).

% max_bot
tff(fact_4211_max__bot,axiom,
    ! [Xc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,bot_bot_nat),Xc) = Xc ).

% max_bot
tff(fact_4212_pos__mult__pos__ge,axiom,
    ! [Xc: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Nb),one_one_int)),aa_int_int(times_times_int(Nb),Xc)) ) ) ).

% pos_mult_pos_ge
tff(fact_4213_dual__order_Orefl,axiom,
    ! [A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A2),A2) ).

% dual_order.refl
tff(fact_4214_dual__order_Orefl,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),A2) ).

% dual_order.refl
tff(fact_4215_dual__order_Orefl,axiom,
    ! [A2: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),A2) ).

% dual_order.refl
tff(fact_4216_dual__order_Orefl,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),A2) ).

% dual_order.refl
tff(fact_4217_dual__order_Orefl,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),A2) ).

% dual_order.refl
tff(fact_4218_order__refl,axiom,
    ! [Xc: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(Xc),Xc) ).

% order_refl
tff(fact_4219_order__refl,axiom,
    ! [Xc: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Xc) ).

% order_refl
tff(fact_4220_order__refl,axiom,
    ! [Xc: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Xc) ).

% order_refl
tff(fact_4221_order__refl,axiom,
    ! [Xc: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Xc) ).

% order_refl
tff(fact_4222_order__refl,axiom,
    ! [Xc: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Xc) ).

% order_refl
tff(fact_4223_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y2: num] :
      ( ( aa_num_num(bit0,X22) = aa_num_num(bit0,Y2) )
    <=> ( X22 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_4224_star__false__left,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(bot_bot_assn),P) = bot_bot_assn ).

% star_false_left
tff(fact_4225_star__false__right,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(P),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_4226_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_assn(pure_assn,(P)) = bot_bot_assn )
    <=> ~ (P) ) ).

% pure_assn_eq_false_iff
tff(fact_4227_pure__false,axiom,
    aa_o_assn(pure_assn,$false) = bot_bot_assn ).

% pure_false
tff(fact_4228_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_4229_ent__false__iff,axiom,
    ! [P: assn] :
      ( entails(P,bot_bot_assn)
    <=> ! [H3: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(P),H3) ) ).

% ent_false_iff
tff(fact_4230_snga__same__false,axiom,
    ! [P3: array_VEBT_VEBTi,Xc: list_VEBT_VEBTi,Ya: list_VEBT_VEBTi] : aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(P3),Xc)),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(P3),Ya)) = bot_bot_assn ).

% snga_same_false
tff(fact_4231_bot__set__def,axiom,
    bot_bot_set_complex = collect_complex(bot_bo8750099748576881071plex_o) ).

% bot_set_def
tff(fact_4232_bot__set__def,axiom,
    bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(bot_bo3516217578726220278_int_o) ).

% bot_set_def
tff(fact_4233_bot__set__def,axiom,
    bot_bot_set_real = collect_real(bot_bot_fun_real_o) ).

% bot_set_def
tff(fact_4234_bot__set__def,axiom,
    bot_bot_set_o = collect_o(bot_bot_fun_o_o) ).

% bot_set_def
tff(fact_4235_bot__set__def,axiom,
    bot_bot_set_nat = collect_nat(bot_bot_fun_nat_o) ).

% bot_set_def
tff(fact_4236_bot__set__def,axiom,
    bot_bot_set_int = collect_int(bot_bot_fun_int_o) ).

% bot_set_def
tff(fact_4237_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_4238_bot__option__def,axiom,
    bot_bot_option_nat = none_nat ).

% bot_option_def
tff(fact_4239_ent__false,axiom,
    ! [P: assn] : entails(bot_bot_assn,P) ).

% ent_false
tff(fact_4240_mod__false,axiom,
    ! [H: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(bot_bot_assn),H) ).

% mod_false
tff(fact_4241_list__assn__aux__ineq__len,axiom,
    ! [L: list_VEBT_VEBT,Li2: list_VEBT_VEBT,A: fun_VE1571388752190420990T_assn] :
      ( ( size_s6755466524823107622T_VEBT(L) != size_s6755466524823107622T_VEBT(Li2) )
     => ( vEBT_L1279224858307276611T_VEBT(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_4242_list__assn__aux__ineq__len,axiom,
    ! [L: list_VEBT_VEBT,Li2: list_real,A: fun_VE6910866152645720856l_assn] :
      ( ( size_s6755466524823107622T_VEBT(L) != size_size_list_real(Li2) )
     => ( vEBT_L5781919052683127133T_real(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_4243_list__assn__aux__ineq__len,axiom,
    ! [L: list_VEBT_VEBT,Li2: list_o,A: fun_VE2104784059010244834o_assn] :
      ( ( size_s6755466524823107622T_VEBT(L) != size_size_list_o(Li2) )
     => ( vEBT_L7489408758114837031VEBT_o(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_4244_list__assn__aux__ineq__len,axiom,
    ! [L: list_VEBT_VEBT,Li2: list_nat,A: fun_VE4931415397359990460t_assn] :
      ( ( size_s6755466524823107622T_VEBT(L) != size_size_list_nat(Li2) )
     => ( vEBT_L8296926524756676353BT_nat(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_4245_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_VEBT_VEBT,A: fun_re8050108611700907676T_assn] :
      ( ( size_size_list_real(L) != size_s6755466524823107622T_VEBT(Li2) )
     => ( vEBT_L4595930785310033027T_VEBT(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_4246_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_real,A: fun_re6303261851654292474l_assn] :
      ( ( size_size_list_real(L) != size_size_list_real(Li2) )
     => ( vEBT_L1930518968523514909l_real(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_4247_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_o,A: fun_real_fun_o_assn] :
      ( ( size_size_list_real(L) != size_size_list_o(Li2) )
     => ( vEBT_L6234343332106409831real_o(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_4248_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_nat,A: fun_re574865750389657758t_assn] :
      ( ( size_size_list_real(L) != size_size_list_nat(Li2) )
     => ( vEBT_L1446010312343316929al_nat(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_4249_list__assn__aux__ineq__len,axiom,
    ! [L: list_real,Li2: list_VEBT_VEBTi,A: fun_re560840401387337987i_assn] :
      ( ( size_size_list_real(L) != size_s7982070591426661849_VEBTi(Li2) )
     => ( aa_lis2508312872377092907i_assn(vEBT_L9060850011106065574_VEBTi(A,L),Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_4250_list__assn__aux__ineq__len,axiom,
    ! [L: list_o,Li2: list_VEBT_VEBT,A: fun_o_4353168661238401050T_assn] :
      ( ( size_size_list_o(L) != size_s6755466524823107622T_VEBT(Li2) )
     => ( vEBT_L1750719106661372127T_VEBT(A,L,Li2) = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
tff(fact_4251_order__antisym__conv,axiom,
    ! [Ya: set_nat,Xc: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Ya),Xc)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% order_antisym_conv
tff(fact_4252_order__antisym__conv,axiom,
    ! [Ya: rat,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% order_antisym_conv
tff(fact_4253_order__antisym__conv,axiom,
    ! [Ya: num,Xc: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% order_antisym_conv
tff(fact_4254_order__antisym__conv,axiom,
    ! [Ya: nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% order_antisym_conv
tff(fact_4255_order__antisym__conv,axiom,
    ! [Ya: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% order_antisym_conv
tff(fact_4256_linorder__le__cases,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc) ) ).

% linorder_le_cases
tff(fact_4257_linorder__le__cases,axiom,
    ! [Xc: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc) ) ).

% linorder_le_cases
tff(fact_4258_linorder__le__cases,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc) ) ).

% linorder_le_cases
tff(fact_4259_linorder__le__cases,axiom,
    ! [Xc: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc) ) ).

% linorder_le_cases
tff(fact_4260_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( ( aa_rat_rat(F2,B2) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F2,A2)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_4261_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( ( aa_rat_num(F2,B2) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F2,A2)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_4262_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( ( aa_rat_nat(F2,B2) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F2,A2)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_4263_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( ( aa_rat_int(F2,B2) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F2,A2)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_4264_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F2: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( ( aa_num_rat(F2,B2) = C2 )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F2,A2)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_4265_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F2: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( ( aa_num_num(F2,B2) = C2 )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F2,A2)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_4266_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F2: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( ( aa_num_nat(F2,B2) = C2 )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F2,A2)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_4267_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F2: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( ( aa_num_int(F2,B2) = C2 )
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F2,X3)),aa_num_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F2,A2)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_4268_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F2: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( ( aa_nat_rat(F2,B2) = C2 )
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,A2)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_4269_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F2: fun_nat_num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( ( aa_nat_num(F2,B2) = C2 )
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F2,X3)),aa_nat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F2,A2)),C2) ) ) ) ).

% ord_le_eq_subst
tff(fact_4270_ord__eq__le__subst,axiom,
    ! [A2: rat,F2: fun_rat_rat,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_rat(F2,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(F2,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_4271_ord__eq__le__subst,axiom,
    ! [A2: num,F2: fun_rat_num,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_num(F2,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_rat_num(F2,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_4272_ord__eq__le__subst,axiom,
    ! [A2: nat,F2: fun_rat_nat,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_nat(F2,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_rat_nat(F2,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_4273_ord__eq__le__subst,axiom,
    ! [A2: int,F2: fun_rat_int,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_int(F2,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_rat_int(F2,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_4274_ord__eq__le__subst,axiom,
    ! [A2: rat,F2: fun_num_rat,B2: num,C2: num] :
      ( ( A2 = aa_num_rat(F2,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_num_rat(F2,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_4275_ord__eq__le__subst,axiom,
    ! [A2: num,F2: fun_num_num,B2: num,C2: num] :
      ( ( A2 = aa_num_num(F2,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_num_num(F2,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_4276_ord__eq__le__subst,axiom,
    ! [A2: nat,F2: fun_num_nat,B2: num,C2: num] :
      ( ( A2 = aa_num_nat(F2,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_num_nat(F2,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_4277_ord__eq__le__subst,axiom,
    ! [A2: int,F2: fun_num_int,B2: num,C2: num] :
      ( ( A2 = aa_num_int(F2,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F2,X3)),aa_num_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_num_int(F2,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_4278_ord__eq__le__subst,axiom,
    ! [A2: rat,F2: fun_nat_rat,B2: nat,C2: nat] :
      ( ( A2 = aa_nat_rat(F2,B2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_nat_rat(F2,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_4279_ord__eq__le__subst,axiom,
    ! [A2: num,F2: fun_nat_num,B2: nat,C2: nat] :
      ( ( A2 = aa_nat_num(F2,B2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F2,X3)),aa_nat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_nat_num(F2,C2)) ) ) ) ).

% ord_eq_le_subst
tff(fact_4280_linorder__linear,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc) ) ).

% linorder_linear
tff(fact_4281_linorder__linear,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc) ) ).

% linorder_linear
tff(fact_4282_linorder__linear,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc) ) ).

% linorder_linear
tff(fact_4283_linorder__linear,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc) ) ).

% linorder_linear
tff(fact_4284_verit__la__disequality,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ).

% verit_la_disequality
tff(fact_4285_verit__la__disequality,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 = B2 )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2) ) ).

% verit_la_disequality
tff(fact_4286_verit__la__disequality,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2) ) ).

% verit_la_disequality
tff(fact_4287_verit__la__disequality,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ).

% verit_la_disequality
tff(fact_4288_order__eq__refl,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( ( Xc = Ya )
     => aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya) ) ).

% order_eq_refl
tff(fact_4289_order__eq__refl,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ( Xc = Ya )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya) ) ).

% order_eq_refl
tff(fact_4290_order__eq__refl,axiom,
    ! [Xc: num,Ya: num] :
      ( ( Xc = Ya )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya) ) ).

% order_eq_refl
tff(fact_4291_order__eq__refl,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( Xc = Ya )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya) ) ).

% order_eq_refl
tff(fact_4292_order__eq__refl,axiom,
    ! [Xc: int,Ya: int] :
      ( ( Xc = Ya )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya) ) ).

% order_eq_refl
tff(fact_4293_order__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F2,A2)),C2) ) ) ) ).

% order_subst2
tff(fact_4294_order__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F2,A2)),C2) ) ) ) ).

% order_subst2
tff(fact_4295_order__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F2,A2)),C2) ) ) ) ).

% order_subst2
tff(fact_4296_order__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F2,A2)),C2) ) ) ) ).

% order_subst2
tff(fact_4297_order__subst2,axiom,
    ! [A2: num,B2: num,F2: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F2,B2)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F2,A2)),C2) ) ) ) ).

% order_subst2
tff(fact_4298_order__subst2,axiom,
    ! [A2: num,B2: num,F2: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F2,B2)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F2,A2)),C2) ) ) ) ).

% order_subst2
tff(fact_4299_order__subst2,axiom,
    ! [A2: num,B2: num,F2: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F2,B2)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F2,A2)),C2) ) ) ) ).

% order_subst2
tff(fact_4300_order__subst2,axiom,
    ! [A2: num,B2: num,F2: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F2,B2)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F2,X3)),aa_num_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F2,A2)),C2) ) ) ) ).

% order_subst2
tff(fact_4301_order__subst2,axiom,
    ! [A2: nat,B2: nat,F2: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,B2)),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,A2)),C2) ) ) ) ).

% order_subst2
tff(fact_4302_order__subst2,axiom,
    ! [A2: nat,B2: nat,F2: fun_nat_num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F2,B2)),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F2,X3)),aa_nat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F2,A2)),C2) ) ) ) ).

% order_subst2
tff(fact_4303_order__subst1,axiom,
    ! [A2: rat,F2: fun_rat_rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(F2,C2)) ) ) ) ).

% order_subst1
tff(fact_4304_order__subst1,axiom,
    ! [A2: rat,F2: fun_num_rat,B2: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_num_rat(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_num_rat(F2,C2)) ) ) ) ).

% order_subst1
tff(fact_4305_order__subst1,axiom,
    ! [A2: rat,F2: fun_nat_rat,B2: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_nat_rat(F2,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_nat_rat(F2,C2)) ) ) ) ).

% order_subst1
tff(fact_4306_order__subst1,axiom,
    ! [A2: rat,F2: fun_int_rat,B2: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_int_rat(F2,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F2,X3)),aa_int_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_int_rat(F2,C2)) ) ) ) ).

% order_subst1
tff(fact_4307_order__subst1,axiom,
    ! [A2: num,F2: fun_rat_num,B2: rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_rat_num(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_rat_num(F2,C2)) ) ) ) ).

% order_subst1
tff(fact_4308_order__subst1,axiom,
    ! [A2: num,F2: fun_num_num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_num_num(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_num_num(F2,C2)) ) ) ) ).

% order_subst1
tff(fact_4309_order__subst1,axiom,
    ! [A2: num,F2: fun_nat_num,B2: nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_nat_num(F2,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F2,X3)),aa_nat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_nat_num(F2,C2)) ) ) ) ).

% order_subst1
tff(fact_4310_order__subst1,axiom,
    ! [A2: num,F2: fun_int_num,B2: int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_int_num(F2,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F2,X3)),aa_int_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_int_num(F2,C2)) ) ) ) ).

% order_subst1
tff(fact_4311_order__subst1,axiom,
    ! [A2: nat,F2: fun_rat_nat,B2: rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_rat_nat(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_rat_nat(F2,C2)) ) ) ) ).

% order_subst1
tff(fact_4312_order__subst1,axiom,
    ! [A2: nat,F2: fun_num_nat,B2: num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_num_nat(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_num_nat(F2,C2)) ) ) ) ).

% order_subst1
tff(fact_4313_Orderings_Oorder__eq__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        & aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_4314_Orderings_Oorder__eq__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_4315_Orderings_Oorder__eq__iff,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 = B2 )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_4316_Orderings_Oorder__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_4317_Orderings_Oorder__eq__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ) ).

% Orderings.order_eq_iff
tff(fact_4318_antisym,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_4319_antisym,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_4320_antisym,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_4321_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_4322_antisym,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
       => ( A2 = B2 ) ) ) ).

% antisym
tff(fact_4323_dual__order_Otrans,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C2),B2)
       => aa_set_nat_o(ord_less_eq_set_nat(C2),A2) ) ) ).

% dual_order.trans
tff(fact_4324_dual__order_Otrans,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A2) ) ) ).

% dual_order.trans
tff(fact_4325_dual__order_Otrans,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A2) ) ) ).

% dual_order.trans
tff(fact_4326_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A2) ) ) ).

% dual_order.trans
tff(fact_4327_dual__order_Otrans,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A2) ) ) ).

% dual_order.trans
tff(fact_4328_dual__order_Oantisym,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_4329_dual__order_Oantisym,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_4330_dual__order_Oantisym,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_4331_dual__order_Oantisym,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_4332_dual__order_Oantisym,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
tff(fact_4333_dual__order_Oeq__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
        & aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ) ).

% dual_order.eq_iff
tff(fact_4334_dual__order_Oeq__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% dual_order.eq_iff
tff(fact_4335_dual__order_Oeq__iff,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 = B2 )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2) ) ) ).

% dual_order.eq_iff
tff(fact_4336_dual__order_Oeq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ) ).

% dual_order.eq_iff
tff(fact_4337_dual__order_Oeq__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ).

% dual_order.eq_iff
tff(fact_4338_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A2: rat,B2: rat] :
      ( ! [A3: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A3),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B3) )
     => ( ! [A3: rat,B3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(P,B3),A3)
           => aa_rat_o(aa_rat_fun_rat_o(P,A3),B3) )
       => aa_rat_o(aa_rat_fun_rat_o(P,A2),B2) ) ) ).

% linorder_wlog
tff(fact_4339_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_o,A2: num,B2: num] :
      ( ! [A3: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A3),B3)
         => aa_num_o(aa_num_fun_num_o(P,A3),B3) )
     => ( ! [A3: num,B3: num] :
            ( aa_num_o(aa_num_fun_num_o(P,B3),A3)
           => aa_num_o(aa_num_fun_num_o(P,A3),B3) )
       => aa_num_o(aa_num_fun_num_o(P,A2),B2) ) ) ).

% linorder_wlog
tff(fact_4340_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
     => ( ! [A3: nat,B3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(P,B3),A3)
           => aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
       => aa_nat_o(aa_nat_fun_nat_o(P,A2),B2) ) ) ).

% linorder_wlog
tff(fact_4341_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,A2: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(P,A3),B3) )
     => ( ! [A3: int,B3: int] :
            ( aa_int_o(aa_int_fun_int_o(P,B3),A3)
           => aa_int_o(aa_int_fun_int_o(P,A3),B3) )
       => aa_int_o(aa_int_fun_int_o(P,A2),B2) ) ) ).

% linorder_wlog
tff(fact_4342_order__trans,axiom,
    ! [Xc: set_nat,Ya: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Ya),Z)
       => aa_set_nat_o(ord_less_eq_set_nat(Xc),Z) ) ) ).

% order_trans
tff(fact_4343_order__trans,axiom,
    ! [Xc: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Z) ) ) ).

% order_trans
tff(fact_4344_order__trans,axiom,
    ! [Xc: num,Ya: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Z) ) ) ).

% order_trans
tff(fact_4345_order__trans,axiom,
    ! [Xc: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Z) ) ) ).

% order_trans
tff(fact_4346_order__trans,axiom,
    ! [Xc: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Z) ) ) ).

% order_trans
tff(fact_4347_order_Otrans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C2)
       => aa_set_nat_o(ord_less_eq_set_nat(A2),C2) ) ) ).

% order.trans
tff(fact_4348_order_Otrans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),C2) ) ) ).

% order.trans
tff(fact_4349_order_Otrans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),C2) ) ) ).

% order.trans
tff(fact_4350_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),C2) ) ) ).

% order.trans
tff(fact_4351_order_Otrans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),C2) ) ) ).

% order.trans
tff(fact_4352_order__antisym,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Ya),Xc)
       => ( Xc = Ya ) ) ) ).

% order_antisym
tff(fact_4353_order__antisym,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc)
       => ( Xc = Ya ) ) ) ).

% order_antisym
tff(fact_4354_order__antisym,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc)
       => ( Xc = Ya ) ) ) ).

% order_antisym
tff(fact_4355_order__antisym,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc)
       => ( Xc = Ya ) ) ) ).

% order_antisym
tff(fact_4356_order__antisym,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc)
       => ( Xc = Ya ) ) ) ).

% order_antisym
tff(fact_4357_ord__le__eq__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( ( B2 = C2 )
       => aa_set_nat_o(ord_less_eq_set_nat(A2),C2) ) ) ).

% ord_le_eq_trans
tff(fact_4358_ord__le__eq__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( ( B2 = C2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),C2) ) ) ).

% ord_le_eq_trans
tff(fact_4359_ord__le__eq__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( ( B2 = C2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),C2) ) ) ).

% ord_le_eq_trans
tff(fact_4360_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( ( B2 = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),C2) ) ) ).

% ord_le_eq_trans
tff(fact_4361_ord__le__eq__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( ( B2 = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),C2) ) ) ).

% ord_le_eq_trans
tff(fact_4362_ord__eq__le__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( A2 = B2 )
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C2)
       => aa_set_nat_o(ord_less_eq_set_nat(A2),C2) ) ) ).

% ord_eq_le_trans
tff(fact_4363_ord__eq__le__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2 = B2 )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),C2) ) ) ).

% ord_eq_le_trans
tff(fact_4364_ord__eq__le__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( A2 = B2 )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),C2) ) ) ).

% ord_eq_le_trans
tff(fact_4365_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 = B2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),C2) ) ) ).

% ord_eq_le_trans
tff(fact_4366_ord__eq__le__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 = B2 )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),C2) ) ) ).

% ord_eq_le_trans
tff(fact_4367_order__class_Oorder__eq__iff,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( ( Xc = Ya )
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
        & aa_set_nat_o(ord_less_eq_set_nat(Ya),Xc) ) ) ).

% order_class.order_eq_iff
tff(fact_4368_order__class_Oorder__eq__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ( Xc = Ya )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc) ) ) ).

% order_class.order_eq_iff
tff(fact_4369_order__class_Oorder__eq__iff,axiom,
    ! [Xc: num,Ya: num] :
      ( ( Xc = Ya )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc) ) ) ).

% order_class.order_eq_iff
tff(fact_4370_order__class_Oorder__eq__iff,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( Xc = Ya )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc) ) ) ).

% order_class.order_eq_iff
tff(fact_4371_order__class_Oorder__eq__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( ( Xc = Ya )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc) ) ) ).

% order_class.order_eq_iff
tff(fact_4372_le__cases3,axiom,
    ! [Xc: rat,Ya: rat,Z: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Z) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Z)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Ya) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Ya)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xc) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xc)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya) ) ) ) ) ) ) ).

% le_cases3
tff(fact_4373_le__cases3,axiom,
    ! [Xc: num,Ya: num,Z: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Z) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Z) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Z)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Ya) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Ya)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Z)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xc) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xc)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya) ) ) ) ) ) ) ).

% le_cases3
tff(fact_4374_le__cases3,axiom,
    ! [Xc: nat,Ya: nat,Z: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Z) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Z) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Z)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Ya) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Ya)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Z)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xc) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xc)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya) ) ) ) ) ) ) ).

% le_cases3
tff(fact_4375_le__cases3,axiom,
    ! [Xc: int,Ya: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Z) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Z) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Z)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Ya) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Ya)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Z)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xc) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xc)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya) ) ) ) ) ) ) ).

% le_cases3
tff(fact_4376_nle__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
        & ( B2 != A2 ) ) ) ).

% nle_le
tff(fact_4377_nle__le,axiom,
    ! [A2: num,B2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
        & ( B2 != A2 ) ) ) ).

% nle_le
tff(fact_4378_nle__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
        & ( B2 != A2 ) ) ) ).

% nle_le
tff(fact_4379_nle__le,axiom,
    ! [A2: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
        & ( B2 != A2 ) ) ) ).

% nle_le
tff(fact_4380_verit__comp__simplify1_I2_J,axiom,
    ! [A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(A2),A2) ).

% verit_comp_simplify1(2)
tff(fact_4381_verit__comp__simplify1_I2_J,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),A2) ).

% verit_comp_simplify1(2)
tff(fact_4382_verit__comp__simplify1_I2_J,axiom,
    ! [A2: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),A2) ).

% verit_comp_simplify1(2)
tff(fact_4383_verit__comp__simplify1_I2_J,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),A2) ).

% verit_comp_simplify1(2)
tff(fact_4384_verit__comp__simplify1_I2_J,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),A2) ).

% verit_comp_simplify1(2)
tff(fact_4385_verit__comp__simplify1_I1_J,axiom,
    ! [A2: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A2),A2) ).

% verit_comp_simplify1(1)
tff(fact_4386_verit__comp__simplify1_I1_J,axiom,
    ! [A2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),A2) ).

% verit_comp_simplify1(1)
tff(fact_4387_verit__comp__simplify1_I1_J,axiom,
    ! [A2: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A2),A2) ).

% verit_comp_simplify1(1)
tff(fact_4388_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),A2) ).

% verit_comp_simplify1(1)
tff(fact_4389_verit__comp__simplify1_I1_J,axiom,
    ! [A2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),A2) ).

% verit_comp_simplify1(1)
tff(fact_4390_lt__ex,axiom,
    ! [Xc: real] :
    ? [Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xc) ).

% lt_ex
tff(fact_4391_lt__ex,axiom,
    ! [Xc: rat] :
    ? [Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xc) ).

% lt_ex
tff(fact_4392_lt__ex,axiom,
    ! [Xc: int] :
    ? [Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xc) ).

% lt_ex
tff(fact_4393_gt__ex,axiom,
    ! [Xc: real] :
    ? [X_12: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),X_12) ).

% gt_ex
tff(fact_4394_gt__ex,axiom,
    ! [Xc: rat] :
    ? [X_12: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),X_12) ).

% gt_ex
tff(fact_4395_gt__ex,axiom,
    ! [Xc: nat] :
    ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),X_12) ).

% gt_ex
tff(fact_4396_gt__ex,axiom,
    ! [Xc: int] :
    ? [X_12: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),X_12) ).

% gt_ex
tff(fact_4397_dense,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),Ya) ) ) ).

% dense
tff(fact_4398_dense,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ? [Z2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Z2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),Ya) ) ) ).

% dense
tff(fact_4399_less__imp__neq,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ( Xc != Ya ) ) ).

% less_imp_neq
tff(fact_4400_less__imp__neq,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ( Xc != Ya ) ) ).

% less_imp_neq
tff(fact_4401_less__imp__neq,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => ( Xc != Ya ) ) ).

% less_imp_neq
tff(fact_4402_less__imp__neq,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => ( Xc != Ya ) ) ).

% less_imp_neq
tff(fact_4403_less__imp__neq,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => ( Xc != Ya ) ) ).

% less_imp_neq
tff(fact_4404_order_Oasym,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ).

% order.asym
tff(fact_4405_order_Oasym,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ).

% order.asym
tff(fact_4406_order_Oasym,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2) ) ).

% order.asym
tff(fact_4407_order_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2) ) ).

% order.asym
tff(fact_4408_order_Oasym,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ).

% order.asym
tff(fact_4409_ord__eq__less__trans,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( A2 = B2 )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),C2) ) ) ).

% ord_eq_less_trans
tff(fact_4410_ord__eq__less__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2 = B2 )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),C2) ) ) ).

% ord_eq_less_trans
tff(fact_4411_ord__eq__less__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( A2 = B2 )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A2),C2) ) ) ).

% ord_eq_less_trans
tff(fact_4412_ord__eq__less__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 = B2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),C2) ) ) ).

% ord_eq_less_trans
tff(fact_4413_ord__eq__less__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 = B2 )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),C2) ) ) ).

% ord_eq_less_trans
tff(fact_4414_ord__less__eq__trans,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ( B2 = C2 )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),C2) ) ) ).

% ord_less_eq_trans
tff(fact_4415_ord__less__eq__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( ( B2 = C2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),C2) ) ) ).

% ord_less_eq_trans
tff(fact_4416_ord__less__eq__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => ( ( B2 = C2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A2),C2) ) ) ).

% ord_less_eq_trans
tff(fact_4417_ord__less__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( ( B2 = C2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),C2) ) ) ).

% ord_less_eq_trans
tff(fact_4418_ord__less__eq__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( ( B2 = C2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),C2) ) ) ).

% ord_less_eq_trans
tff(fact_4419_less__induct,axiom,
    ! [P: fun_nat_o,A2: nat] :
      ( ! [X3: nat] :
          ( ! [Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y3),X3)
             => aa_nat_o(P,Y3) )
         => aa_nat_o(P,X3) )
     => aa_nat_o(P,A2) ) ).

% less_induct
tff(fact_4420_antisym__conv3,axiom,
    ! [Ya: real,Xc: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv3
tff(fact_4421_antisym__conv3,axiom,
    ! [Ya: rat,Xc: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv3
tff(fact_4422_antisym__conv3,axiom,
    ! [Ya: num,Xc: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv3
tff(fact_4423_antisym__conv3,axiom,
    ! [Ya: nat,Xc: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv3
tff(fact_4424_antisym__conv3,axiom,
    ! [Ya: int,Xc: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv3
tff(fact_4425_linorder__cases,axiom,
    ! [Xc: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ( ( Xc != Ya )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ) ).

% linorder_cases
tff(fact_4426_linorder__cases,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ( ( Xc != Ya )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc) ) ) ).

% linorder_cases
tff(fact_4427_linorder__cases,axiom,
    ! [Xc: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => ( ( Xc != Ya )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc) ) ) ).

% linorder_cases
tff(fact_4428_linorder__cases,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => ( ( Xc != Ya )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc) ) ) ).

% linorder_cases
tff(fact_4429_linorder__cases,axiom,
    ! [Xc: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => ( ( Xc != Ya )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc) ) ) ).

% linorder_cases
tff(fact_4430_dual__order_Oasym,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% dual_order.asym
tff(fact_4431_dual__order_Oasym,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% dual_order.asym
tff(fact_4432_dual__order_Oasym,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ).

% dual_order.asym
tff(fact_4433_dual__order_Oasym,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ).

% dual_order.asym
tff(fact_4434_dual__order_Oasym,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% dual_order.asym
tff(fact_4435_dual__order_Oirrefl,axiom,
    ! [A2: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A2),A2) ).

% dual_order.irrefl
tff(fact_4436_dual__order_Oirrefl,axiom,
    ! [A2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),A2) ).

% dual_order.irrefl
tff(fact_4437_dual__order_Oirrefl,axiom,
    ! [A2: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A2),A2) ).

% dual_order.irrefl
tff(fact_4438_dual__order_Oirrefl,axiom,
    ! [A2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),A2) ).

% dual_order.irrefl
tff(fact_4439_dual__order_Oirrefl,axiom,
    ! [A2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),A2) ).

% dual_order.irrefl
tff(fact_4440_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_1: nat] : aa_nat_o(P,X_1)
    <=> ? [N6: nat] :
          ( aa_nat_o(P,N6)
          & ! [M8: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M8),N6)
             => ~ aa_nat_o(P,M8) ) ) ) ).

% exists_least_iff
tff(fact_4441_linorder__less__wlog,axiom,
    ! [P: fun_real_fun_real_o,A2: real,B2: real] :
      ( ! [A3: real,B3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,A3),B3)
         => aa_real_o(aa_real_fun_real_o(P,A3),B3) )
     => ( ! [A3: real] : aa_real_o(aa_real_fun_real_o(P,A3),A3)
       => ( ! [A3: real,B3: real] :
              ( aa_real_o(aa_real_fun_real_o(P,B3),A3)
             => aa_real_o(aa_real_fun_real_o(P,A3),B3) )
         => aa_real_o(aa_real_fun_real_o(P,A2),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_4442_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A2: rat,B2: rat] :
      ( ! [A3: rat,B3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A3),B3)
         => aa_rat_o(aa_rat_fun_rat_o(P,A3),B3) )
     => ( ! [A3: rat] : aa_rat_o(aa_rat_fun_rat_o(P,A3),A3)
       => ( ! [A3: rat,B3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(P,B3),A3)
             => aa_rat_o(aa_rat_fun_rat_o(P,A3),B3) )
         => aa_rat_o(aa_rat_fun_rat_o(P,A2),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_4443_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_o,A2: num,B2: num] :
      ( ! [A3: num,B3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,A3),B3)
         => aa_num_o(aa_num_fun_num_o(P,A3),B3) )
     => ( ! [A3: num] : aa_num_o(aa_num_fun_num_o(P,A3),A3)
       => ( ! [A3: num,B3: num] :
              ( aa_num_o(aa_num_fun_num_o(P,B3),A3)
             => aa_num_o(aa_num_fun_num_o(P,A3),B3) )
         => aa_num_o(aa_num_fun_num_o(P,A2),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_4444_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
         => aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),A3)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,B3),A3)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),B3) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A2),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_4445_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,A2: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
         => aa_int_o(aa_int_fun_int_o(P,A3),B3) )
     => ( ! [A3: int] : aa_int_o(aa_int_fun_int_o(P,A3),A3)
       => ( ! [A3: int,B3: int] :
              ( aa_int_o(aa_int_fun_int_o(P,B3),A3)
             => aa_int_o(aa_int_fun_int_o(P,A3),B3) )
         => aa_int_o(aa_int_fun_int_o(P,A2),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_4446_order_Ostrict__trans,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),C2) ) ) ).

% order.strict_trans
tff(fact_4447_order_Ostrict__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),C2) ) ) ).

% order.strict_trans
tff(fact_4448_order_Ostrict__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A2),C2) ) ) ).

% order.strict_trans
tff(fact_4449_order_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),C2) ) ) ).

% order.strict_trans
tff(fact_4450_order_Ostrict__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),C2) ) ) ).

% order.strict_trans
tff(fact_4451_not__less__iff__gr__or__eq,axiom,
    ! [Xc: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc)
        | ( Xc = Ya ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4452_not__less__iff__gr__or__eq,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc)
        | ( Xc = Ya ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4453_not__less__iff__gr__or__eq,axiom,
    ! [Xc: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc)
        | ( Xc = Ya ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4454_not__less__iff__gr__or__eq,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc)
        | ( Xc = Ya ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4455_not__less__iff__gr__or__eq,axiom,
    ! [Xc: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc)
        | ( Xc = Ya ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4456_dual__order_Ostrict__trans,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A2) ) ) ).

% dual_order.strict_trans
tff(fact_4457_dual__order_Ostrict__trans,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A2) ) ) ).

% dual_order.strict_trans
tff(fact_4458_dual__order_Ostrict__trans,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A2) ) ) ).

% dual_order.strict_trans
tff(fact_4459_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A2) ) ) ).

% dual_order.strict_trans
tff(fact_4460_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A2) ) ) ).

% dual_order.strict_trans
tff(fact_4461_order_Ostrict__implies__not__eq,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_4462_order_Ostrict__implies__not__eq,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_4463_order_Ostrict__implies__not__eq,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_4464_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_4465_order_Ostrict__implies__not__eq,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_4466_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4467_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4468_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4469_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4470_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4471_linorder__neqE,axiom,
    ! [Xc: real,Ya: real] :
      ( ( Xc != Ya )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ) ).

% linorder_neqE
tff(fact_4472_linorder__neqE,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ( Xc != Ya )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc) ) ) ).

% linorder_neqE
tff(fact_4473_linorder__neqE,axiom,
    ! [Xc: num,Ya: num] :
      ( ( Xc != Ya )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc) ) ) ).

% linorder_neqE
tff(fact_4474_linorder__neqE,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( Xc != Ya )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc) ) ) ).

% linorder_neqE
tff(fact_4475_linorder__neqE,axiom,
    ! [Xc: int,Ya: int] :
      ( ( Xc != Ya )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc) ) ) ).

% linorder_neqE
tff(fact_4476_order__less__asym,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ).

% order_less_asym
tff(fact_4477_order__less__asym,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc) ) ).

% order_less_asym
tff(fact_4478_order__less__asym,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc) ) ).

% order_less_asym
tff(fact_4479_order__less__asym,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc) ) ).

% order_less_asym
tff(fact_4480_order__less__asym,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc) ) ).

% order_less_asym
tff(fact_4481_linorder__neq__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( ( Xc != Ya )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ) ).

% linorder_neq_iff
tff(fact_4482_linorder__neq__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ( Xc != Ya )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc) ) ) ).

% linorder_neq_iff
tff(fact_4483_linorder__neq__iff,axiom,
    ! [Xc: num,Ya: num] :
      ( ( Xc != Ya )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc) ) ) ).

% linorder_neq_iff
tff(fact_4484_linorder__neq__iff,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( Xc != Ya )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc) ) ) ).

% linorder_neq_iff
tff(fact_4485_linorder__neq__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( ( Xc != Ya )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc) ) ) ).

% linorder_neq_iff
tff(fact_4486_order__less__asym_H,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2) ) ).

% order_less_asym'
tff(fact_4487_order__less__asym_H,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2) ) ).

% order_less_asym'
tff(fact_4488_order__less__asym_H,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2) ) ).

% order_less_asym'
tff(fact_4489_order__less__asym_H,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2) ) ).

% order_less_asym'
tff(fact_4490_order__less__asym_H,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2) ) ).

% order_less_asym'
tff(fact_4491_order__less__trans,axiom,
    ! [Xc: real,Ya: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Z) ) ) ).

% order_less_trans
tff(fact_4492_order__less__trans,axiom,
    ! [Xc: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Z) ) ) ).

% order_less_trans
tff(fact_4493_order__less__trans,axiom,
    ! [Xc: num,Ya: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Z) ) ) ).

% order_less_trans
tff(fact_4494_order__less__trans,axiom,
    ! [Xc: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Z) ) ) ).

% order_less_trans
tff(fact_4495_order__less__trans,axiom,
    ! [Xc: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Z) ) ) ).

% order_less_trans
tff(fact_4496_ord__eq__less__subst,axiom,
    ! [A2: real,F2: fun_real_real,B2: real,C2: real] :
      ( ( A2 = aa_real_real(F2,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(F2,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4497_ord__eq__less__subst,axiom,
    ! [A2: rat,F2: fun_real_rat,B2: real,C2: real] :
      ( ( A2 = aa_real_rat(F2,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_real_rat(F2,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4498_ord__eq__less__subst,axiom,
    ! [A2: num,F2: fun_real_num,B2: real,C2: real] :
      ( ( A2 = aa_real_num(F2,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F2,X3)),aa_real_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A2),aa_real_num(F2,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4499_ord__eq__less__subst,axiom,
    ! [A2: nat,F2: fun_real_nat,B2: real,C2: real] :
      ( ( A2 = aa_real_nat(F2,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F2,X3)),aa_real_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_real_nat(F2,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4500_ord__eq__less__subst,axiom,
    ! [A2: int,F2: fun_real_int,B2: real,C2: real] :
      ( ( A2 = aa_real_int(F2,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F2,X3)),aa_real_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_real_int(F2,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4501_ord__eq__less__subst,axiom,
    ! [A2: real,F2: fun_rat_real,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_real(F2,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_rat_real(F2,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4502_ord__eq__less__subst,axiom,
    ! [A2: rat,F2: fun_rat_rat,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_rat(F2,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(F2,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4503_ord__eq__less__subst,axiom,
    ! [A2: num,F2: fun_rat_num,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_num(F2,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A2),aa_rat_num(F2,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4504_ord__eq__less__subst,axiom,
    ! [A2: nat,F2: fun_rat_nat,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_nat(F2,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_rat_nat(F2,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4505_ord__eq__less__subst,axiom,
    ! [A2: int,F2: fun_rat_int,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_int(F2,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_rat_int(F2,C2)) ) ) ) ).

% ord_eq_less_subst
tff(fact_4506_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F2: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ( aa_real_real(F2,B2) = C2 )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,A2)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_4507_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F2: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ( aa_real_rat(F2,B2) = C2 )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F2,A2)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_4508_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F2: fun_real_num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ( aa_real_num(F2,B2) = C2 )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F2,X3)),aa_real_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F2,A2)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_4509_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F2: fun_real_nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ( aa_real_nat(F2,B2) = C2 )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F2,X3)),aa_real_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F2,A2)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_4510_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F2: fun_real_int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ( aa_real_int(F2,B2) = C2 )
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F2,X3)),aa_real_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F2,A2)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_4511_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( ( aa_rat_real(F2,B2) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,A2)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_4512_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( ( aa_rat_rat(F2,B2) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,A2)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_4513_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( ( aa_rat_num(F2,B2) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F2,A2)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_4514_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( ( aa_rat_nat(F2,B2) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F2,A2)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_4515_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( ( aa_rat_int(F2,B2) = C2 )
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F2,A2)),C2) ) ) ) ).

% ord_less_eq_subst
tff(fact_4516_order__less__irrefl,axiom,
    ! [Xc: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Xc) ).

% order_less_irrefl
tff(fact_4517_order__less__irrefl,axiom,
    ! [Xc: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Xc) ).

% order_less_irrefl
tff(fact_4518_order__less__irrefl,axiom,
    ! [Xc: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Xc) ).

% order_less_irrefl
tff(fact_4519_order__less__irrefl,axiom,
    ! [Xc: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Xc) ).

% order_less_irrefl
tff(fact_4520_order__less__irrefl,axiom,
    ! [Xc: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Xc) ).

% order_less_irrefl
tff(fact_4521_order__less__subst1,axiom,
    ! [A2: real,F2: fun_real_real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(F2,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(F2,C2)) ) ) ) ).

% order_less_subst1
tff(fact_4522_order__less__subst1,axiom,
    ! [A2: real,F2: fun_rat_real,B2: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_rat_real(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_rat_real(F2,C2)) ) ) ) ).

% order_less_subst1
tff(fact_4523_order__less__subst1,axiom,
    ! [A2: real,F2: fun_num_real,B2: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_num_real(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F2,X3)),aa_num_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_num_real(F2,C2)) ) ) ) ).

% order_less_subst1
tff(fact_4524_order__less__subst1,axiom,
    ! [A2: real,F2: fun_nat_real,B2: nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_nat_real(F2,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F2,X3)),aa_nat_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_nat_real(F2,C2)) ) ) ) ).

% order_less_subst1
tff(fact_4525_order__less__subst1,axiom,
    ! [A2: real,F2: fun_int_real,B2: int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_int_real(F2,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F2,X3)),aa_int_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_int_real(F2,C2)) ) ) ) ).

% order_less_subst1
tff(fact_4526_order__less__subst1,axiom,
    ! [A2: rat,F2: fun_real_rat,B2: real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_real_rat(F2,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_real_rat(F2,C2)) ) ) ) ).

% order_less_subst1
tff(fact_4527_order__less__subst1,axiom,
    ! [A2: rat,F2: fun_rat_rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(F2,C2)) ) ) ) ).

% order_less_subst1
tff(fact_4528_order__less__subst1,axiom,
    ! [A2: rat,F2: fun_num_rat,B2: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_num_rat(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_num_rat(F2,C2)) ) ) ) ).

% order_less_subst1
tff(fact_4529_order__less__subst1,axiom,
    ! [A2: rat,F2: fun_nat_rat,B2: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_nat_rat(F2,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_nat_rat(F2,C2)) ) ) ) ).

% order_less_subst1
tff(fact_4530_order__less__subst1,axiom,
    ! [A2: rat,F2: fun_int_rat,B2: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_int_rat(F2,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F2,X3)),aa_int_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_int_rat(F2,C2)) ) ) ) ).

% order_less_subst1
tff(fact_4531_order__less__subst2,axiom,
    ! [A2: real,B2: real,F2: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,B2)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,A2)),C2) ) ) ) ).

% order_less_subst2
tff(fact_4532_order__less__subst2,axiom,
    ! [A2: real,B2: real,F2: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F2,B2)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F2,A2)),C2) ) ) ) ).

% order_less_subst2
tff(fact_4533_order__less__subst2,axiom,
    ! [A2: real,B2: real,F2: fun_real_num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F2,B2)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F2,X3)),aa_real_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F2,A2)),C2) ) ) ) ).

% order_less_subst2
tff(fact_4534_order__less__subst2,axiom,
    ! [A2: real,B2: real,F2: fun_real_nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F2,B2)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F2,X3)),aa_real_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F2,A2)),C2) ) ) ) ).

% order_less_subst2
tff(fact_4535_order__less__subst2,axiom,
    ! [A2: real,B2: real,F2: fun_real_int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F2,B2)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F2,X3)),aa_real_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F2,A2)),C2) ) ) ) ).

% order_less_subst2
tff(fact_4536_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,A2)),C2) ) ) ) ).

% order_less_subst2
tff(fact_4537_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,A2)),C2) ) ) ) ).

% order_less_subst2
tff(fact_4538_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F2,A2)),C2) ) ) ) ).

% order_less_subst2
tff(fact_4539_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F2,A2)),C2) ) ) ) ).

% order_less_subst2
tff(fact_4540_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F2,A2)),C2) ) ) ) ).

% order_less_subst2
tff(fact_4541_order__less__not__sym,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ).

% order_less_not_sym
tff(fact_4542_order__less__not__sym,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc) ) ).

% order_less_not_sym
tff(fact_4543_order__less__not__sym,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc) ) ).

% order_less_not_sym
tff(fact_4544_order__less__not__sym,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc) ) ).

% order_less_not_sym
tff(fact_4545_order__less__not__sym,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc) ) ).

% order_less_not_sym
tff(fact_4546_order__less__imp__triv,axiom,
    ! [Xc: real,Ya: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_4547_order__less__imp__triv,axiom,
    ! [Xc: rat,Ya: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_4548_order__less__imp__triv,axiom,
    ! [Xc: num,Ya: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_4549_order__less__imp__triv,axiom,
    ! [Xc: nat,Ya: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_4550_order__less__imp__triv,axiom,
    ! [Xc: int,Ya: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_4551_linorder__less__linear,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
      | ( Xc = Ya )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ).

% linorder_less_linear
tff(fact_4552_linorder__less__linear,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
      | ( Xc = Ya )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc) ) ).

% linorder_less_linear
tff(fact_4553_linorder__less__linear,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
      | ( Xc = Ya )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc) ) ).

% linorder_less_linear
tff(fact_4554_linorder__less__linear,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
      | ( Xc = Ya )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc) ) ).

% linorder_less_linear
tff(fact_4555_linorder__less__linear,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
      | ( Xc = Ya )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc) ) ).

% linorder_less_linear
tff(fact_4556_order__less__imp__not__eq,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ( Xc != Ya ) ) ).

% order_less_imp_not_eq
tff(fact_4557_order__less__imp__not__eq,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ( Xc != Ya ) ) ).

% order_less_imp_not_eq
tff(fact_4558_order__less__imp__not__eq,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => ( Xc != Ya ) ) ).

% order_less_imp_not_eq
tff(fact_4559_order__less__imp__not__eq,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => ( Xc != Ya ) ) ).

% order_less_imp_not_eq
tff(fact_4560_order__less__imp__not__eq,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => ( Xc != Ya ) ) ).

% order_less_imp_not_eq
tff(fact_4561_order__less__imp__not__eq2,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ( Ya != Xc ) ) ).

% order_less_imp_not_eq2
tff(fact_4562_order__less__imp__not__eq2,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ( Ya != Xc ) ) ).

% order_less_imp_not_eq2
tff(fact_4563_order__less__imp__not__eq2,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => ( Ya != Xc ) ) ).

% order_less_imp_not_eq2
tff(fact_4564_order__less__imp__not__eq2,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => ( Ya != Xc ) ) ).

% order_less_imp_not_eq2
tff(fact_4565_order__less__imp__not__eq2,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => ( Ya != Xc ) ) ).

% order_less_imp_not_eq2
tff(fact_4566_order__less__imp__not__less,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ).

% order_less_imp_not_less
tff(fact_4567_order__less__imp__not__less,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc) ) ).

% order_less_imp_not_less
tff(fact_4568_order__less__imp__not__less,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc) ) ).

% order_less_imp_not_less
tff(fact_4569_order__less__imp__not__less,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc) ) ).

% order_less_imp_not_less
tff(fact_4570_order__less__imp__not__less,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc) ) ).

% order_less_imp_not_less
tff(fact_4571_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd2: $o,Ve2: $o,V: option4927543243414619207at_nat,Vaa: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_o_VEBT_VEBT(vEBT_Leaf((Vd2)),(Ve2))),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V,Vaa,Vb2),Vc2)) = bot_bot_assn ).

% vebt_assn_raw.simps(4)
tff(fact_4572_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A4: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B4),A4)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_4573_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A4: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B4),A4)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_4574_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A4: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B4),A4)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_4575_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A4: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B4),A4)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_4576_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A4: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B4),A4)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A4),B4) ) ).

% verit_comp_simplify1(3)
tff(fact_4577_leD,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ).

% leD
tff(fact_4578_leD,axiom,
    ! [Ya: set_nat,Xc: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Ya),Xc)
     => ~ aa_set_nat_o(ord_less_set_nat(Xc),Ya) ) ).

% leD
tff(fact_4579_leD,axiom,
    ! [Ya: rat,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya) ) ).

% leD
tff(fact_4580_leD,axiom,
    ! [Ya: num,Xc: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya) ) ).

% leD
tff(fact_4581_leD,axiom,
    ! [Ya: nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya) ) ).

% leD
tff(fact_4582_leD,axiom,
    ! [Ya: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya) ) ).

% leD
tff(fact_4583_leI,axiom,
    ! [Xc: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc) ) ).

% leI
tff(fact_4584_leI,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc) ) ).

% leI
tff(fact_4585_leI,axiom,
    ! [Xc: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc) ) ).

% leI
tff(fact_4586_leI,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc) ) ).

% leI
tff(fact_4587_leI,axiom,
    ! [Xc: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc) ) ).

% leI
tff(fact_4588_nless__le,axiom,
    ! [A2: real,B2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4589_nless__le,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ~ aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ( ~ aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4590_nless__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4591_nless__le,axiom,
    ! [A2: num,B2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4592_nless__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4593_nless__le,axiom,
    ! [A2: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
        | ( A2 = B2 ) ) ) ).

% nless_le
tff(fact_4594_antisym__conv1,axiom,
    ! [Xc: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv1
tff(fact_4595_antisym__conv1,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( ~ aa_set_nat_o(ord_less_set_nat(Xc),Ya)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv1
tff(fact_4596_antisym__conv1,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv1
tff(fact_4597_antisym__conv1,axiom,
    ! [Xc: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv1
tff(fact_4598_antisym__conv1,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv1
tff(fact_4599_antisym__conv1,axiom,
    ! [Xc: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv1
tff(fact_4600_antisym__conv2,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv2
tff(fact_4601_antisym__conv2,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
     => ( ~ aa_set_nat_o(ord_less_set_nat(Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv2
tff(fact_4602_antisym__conv2,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv2
tff(fact_4603_antisym__conv2,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv2
tff(fact_4604_antisym__conv2,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv2
tff(fact_4605_antisym__conv2,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
      <=> ( Xc = Ya ) ) ) ).

% antisym_conv2
tff(fact_4606_dense__ge,axiom,
    ! [Z: real,Ya: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),X3) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Z) ) ).

% dense_ge
tff(fact_4607_dense__ge,axiom,
    ! [Z: rat,Ya: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),X3) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z) ) ).

% dense_ge
tff(fact_4608_dense__le,axiom,
    ! [Ya: real,Z: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Ya)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Z) ) ).

% dense_le
tff(fact_4609_dense__le,axiom,
    ! [Ya: rat,Z: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Ya)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Z) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z) ) ).

% dense_le
tff(fact_4610_less__le__not__le,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc) ) ) ).

% less_le_not_le
tff(fact_4611_less__le__not__le,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xc),Ya)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(Ya),Xc) ) ) ).

% less_le_not_le
tff(fact_4612_less__le__not__le,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc) ) ) ).

% less_le_not_le
tff(fact_4613_less__le__not__le,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc) ) ) ).

% less_le_not_le
tff(fact_4614_less__le__not__le,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc) ) ) ).

% less_le_not_le
tff(fact_4615_less__le__not__le,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc) ) ) ).

% less_le_not_le
tff(fact_4616_not__le__imp__less,axiom,
    ! [Ya: real,Xc: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ).

% not_le_imp_less
tff(fact_4617_not__le__imp__less,axiom,
    ! [Ya: rat,Xc: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya) ) ).

% not_le_imp_less
tff(fact_4618_not__le__imp__less,axiom,
    ! [Ya: num,Xc: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya) ) ).

% not_le_imp_less
tff(fact_4619_not__le__imp__less,axiom,
    ! [Ya: nat,Xc: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya) ) ).

% not_le_imp_less
tff(fact_4620_not__le__imp__less,axiom,
    ! [Ya: int,Xc: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya) ) ).

% not_le_imp_less
tff(fact_4621_order_Oorder__iff__strict,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4622_order_Oorder__iff__strict,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_set_nat(A2),B2)
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4623_order_Oorder__iff__strict,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4624_order_Oorder__iff__strict,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4625_order_Oorder__iff__strict,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4626_order_Oorder__iff__strict,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
        | ( A2 = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_4627_order_Ostrict__iff__order,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4628_order_Ostrict__iff__order,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4629_order_Ostrict__iff__order,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4630_order_Ostrict__iff__order,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4631_order_Ostrict__iff__order,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4632_order_Ostrict__iff__order,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
        & ( A2 != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_4633_order_Ostrict__trans1,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),C2) ) ) ).

% order.strict_trans1
tff(fact_4634_order_Ostrict__trans1,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_set_nat(B2),C2)
       => aa_set_nat_o(ord_less_set_nat(A2),C2) ) ) ).

% order.strict_trans1
tff(fact_4635_order_Ostrict__trans1,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),C2) ) ) ).

% order.strict_trans1
tff(fact_4636_order_Ostrict__trans1,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A2),C2) ) ) ).

% order.strict_trans1
tff(fact_4637_order_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),C2) ) ) ).

% order.strict_trans1
tff(fact_4638_order_Ostrict__trans1,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),C2) ) ) ).

% order.strict_trans1
tff(fact_4639_order_Ostrict__trans2,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),C2) ) ) ).

% order.strict_trans2
tff(fact_4640_order_Ostrict__trans2,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(B2),C2)
       => aa_set_nat_o(ord_less_set_nat(A2),C2) ) ) ).

% order.strict_trans2
tff(fact_4641_order_Ostrict__trans2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),C2) ) ) ).

% order.strict_trans2
tff(fact_4642_order_Ostrict__trans2,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A2),C2) ) ) ).

% order.strict_trans2
tff(fact_4643_order_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),C2) ) ) ).

% order.strict_trans2
tff(fact_4644_order_Ostrict__trans2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),C2) ) ) ).

% order.strict_trans2
tff(fact_4645_order_Ostrict__iff__not,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ) ).

% order.strict_iff_not
tff(fact_4646_order_Ostrict__iff__not,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ) ).

% order.strict_iff_not
tff(fact_4647_order_Ostrict__iff__not,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ) ).

% order.strict_iff_not
tff(fact_4648_order_Ostrict__iff__not,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2) ) ) ).

% order.strict_iff_not
tff(fact_4649_order_Ostrict__iff__not,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2) ) ) ).

% order.strict_iff_not
tff(fact_4650_order_Ostrict__iff__not,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ) ).

% order.strict_iff_not
tff(fact_4651_dense__ge__bounded,axiom,
    ! [Z: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xc)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Xc)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),W2) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Z) ) ) ).

% dense_ge_bounded
tff(fact_4652_dense__ge__bounded,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xc)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Xc)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),W2) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z) ) ) ).

% dense_ge_bounded
tff(fact_4653_dense__le__bounded,axiom,
    ! [Xc: real,Ya: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Ya)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W2),Z) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Z) ) ) ).

% dense_le_bounded
tff(fact_4654_dense__le__bounded,axiom,
    ! [Xc: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Ya)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W2),Z) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z) ) ) ).

% dense_le_bounded
tff(fact_4655_dual__order_Oorder__iff__strict,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4656_dual__order_Oorder__iff__strict,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
    <=> ( aa_set_nat_o(ord_less_set_nat(B2),A2)
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4657_dual__order_Oorder__iff__strict,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4658_dual__order_Oorder__iff__strict,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4659_dual__order_Oorder__iff__strict,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4660_dual__order_Oorder__iff__strict,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
        | ( A2 = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4661_dual__order_Ostrict__iff__order,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4662_dual__order_Ostrict__iff__order,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B2),A2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4663_dual__order_Ostrict__iff__order,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4664_dual__order_Ostrict__iff__order,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4665_dual__order_Ostrict__iff__order,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4666_dual__order_Ostrict__iff__order,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
        & ( A2 != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4667_dual__order_Ostrict__trans1,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_4668_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
     => ( aa_set_nat_o(ord_less_set_nat(C2),B2)
       => aa_set_nat_o(ord_less_set_nat(C2),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_4669_dual__order_Ostrict__trans1,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_4670_dual__order_Ostrict__trans1,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_4671_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_4672_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A2) ) ) ).

% dual_order.strict_trans1
tff(fact_4673_dual__order_Ostrict__trans2,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_4674_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B2),A2)
     => ( aa_set_nat_o(ord_less_eq_set_nat(C2),B2)
       => aa_set_nat_o(ord_less_set_nat(C2),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_4675_dual__order_Ostrict__trans2,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_4676_dual__order_Ostrict__trans2,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_4677_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_4678_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A2) ) ) ).

% dual_order.strict_trans2
tff(fact_4679_dual__order_Ostrict__iff__not,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_4680_dual__order_Ostrict__iff__not,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B2),A2)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(B2),A2)
        & ~ aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_4681_dual__order_Ostrict__iff__not,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_4682_dual__order_Ostrict__iff__not,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_4683_dual__order_Ostrict__iff__not,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_4684_dual__order_Ostrict__iff__not,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_4685_order_Ostrict__implies__order,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% order.strict_implies_order
tff(fact_4686_order_Ostrict__implies__order,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A2),B2)
     => aa_set_nat_o(ord_less_eq_set_nat(A2),B2) ) ).

% order.strict_implies_order
tff(fact_4687_order_Ostrict__implies__order,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% order.strict_implies_order
tff(fact_4688_order_Ostrict__implies__order,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2) ) ).

% order.strict_implies_order
tff(fact_4689_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2) ) ).

% order.strict_implies_order
tff(fact_4690_order_Ostrict__implies__order,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% order.strict_implies_order
tff(fact_4691_dual__order_Ostrict__implies__order,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A2) ) ).

% dual_order.strict_implies_order
tff(fact_4692_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(B2),A2)
     => aa_set_nat_o(ord_less_eq_set_nat(B2),A2) ) ).

% dual_order.strict_implies_order
tff(fact_4693_dual__order_Ostrict__implies__order,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ).

% dual_order.strict_implies_order
tff(fact_4694_dual__order_Ostrict__implies__order,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2) ) ).

% dual_order.strict_implies_order
tff(fact_4695_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2) ) ).

% dual_order.strict_implies_order
tff(fact_4696_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ).

% dual_order.strict_implies_order
tff(fact_4697_order__le__less,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_less
tff(fact_4698_order__le__less,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
    <=> ( aa_set_nat_o(ord_less_set_nat(Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_less
tff(fact_4699_order__le__less,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_less
tff(fact_4700_order__le__less,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_less
tff(fact_4701_order__le__less,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_less
tff(fact_4702_order__le__less,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_less
tff(fact_4703_order__less__le,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
        & ( Xc != Ya ) ) ) ).

% order_less_le
tff(fact_4704_order__less__le,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xc),Ya)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
        & ( Xc != Ya ) ) ) ).

% order_less_le
tff(fact_4705_order__less__le,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
        & ( Xc != Ya ) ) ) ).

% order_less_le
tff(fact_4706_order__less__le,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
        & ( Xc != Ya ) ) ) ).

% order_less_le
tff(fact_4707_order__less__le,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
        & ( Xc != Ya ) ) ) ).

% order_less_le
tff(fact_4708_order__less__le,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
        & ( Xc != Ya ) ) ) ).

% order_less_le
tff(fact_4709_linorder__not__le,axiom,
    ! [Xc: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ).

% linorder_not_le
tff(fact_4710_linorder__not__le,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc) ) ).

% linorder_not_le
tff(fact_4711_linorder__not__le,axiom,
    ! [Xc: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc) ) ).

% linorder_not_le
tff(fact_4712_linorder__not__le,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc) ) ).

% linorder_not_le
tff(fact_4713_linorder__not__le,axiom,
    ! [Xc: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc) ) ).

% linorder_not_le
tff(fact_4714_linorder__not__less,axiom,
    ! [Xc: real,Ya: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc) ) ).

% linorder_not_less
tff(fact_4715_linorder__not__less,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc) ) ).

% linorder_not_less
tff(fact_4716_linorder__not__less,axiom,
    ! [Xc: num,Ya: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc) ) ).

% linorder_not_less
tff(fact_4717_linorder__not__less,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc) ) ).

% linorder_not_less
tff(fact_4718_linorder__not__less,axiom,
    ! [Xc: int,Ya: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc) ) ).

% linorder_not_less
tff(fact_4719_order__less__imp__le,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ).

% order_less_imp_le
tff(fact_4720_order__less__imp__le,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xc),Ya)
     => aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya) ) ).

% order_less_imp_le
tff(fact_4721_order__less__imp__le,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya) ) ).

% order_less_imp_le
tff(fact_4722_order__less__imp__le,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya) ) ).

% order_less_imp_le
tff(fact_4723_order__less__imp__le,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya) ) ).

% order_less_imp_le
tff(fact_4724_order__less__imp__le,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya) ) ).

% order_less_imp_le
tff(fact_4725_order__le__neq__trans,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( ( A2 != B2 )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% order_le_neq_trans
tff(fact_4726_order__le__neq__trans,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
     => ( ( A2 != B2 )
       => aa_set_nat_o(ord_less_set_nat(A2),B2) ) ) ).

% order_le_neq_trans
tff(fact_4727_order__le__neq__trans,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( ( A2 != B2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% order_le_neq_trans
tff(fact_4728_order__le__neq__trans,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( ( A2 != B2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ) ).

% order_le_neq_trans
tff(fact_4729_order__le__neq__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( ( A2 != B2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ) ).

% order_le_neq_trans
tff(fact_4730_order__le__neq__trans,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( ( A2 != B2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ) ).

% order_le_neq_trans
tff(fact_4731_order__neq__le__trans,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != B2 )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% order_neq_le_trans
tff(fact_4732_order__neq__le__trans,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 != B2 )
     => ( aa_set_nat_o(ord_less_eq_set_nat(A2),B2)
       => aa_set_nat_o(ord_less_set_nat(A2),B2) ) ) ).

% order_neq_le_trans
tff(fact_4733_order__neq__le__trans,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != B2 )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ) ).

% order_neq_le_trans
tff(fact_4734_order__neq__le__trans,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 != B2 )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2) ) ) ).

% order_neq_le_trans
tff(fact_4735_order__neq__le__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2) ) ) ).

% order_neq_le_trans
tff(fact_4736_order__neq__le__trans,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != B2 )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ) ).

% order_neq_le_trans
tff(fact_4737_order__le__less__trans,axiom,
    ! [Xc: real,Ya: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Z) ) ) ).

% order_le_less_trans
tff(fact_4738_order__le__less__trans,axiom,
    ! [Xc: set_nat,Ya: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
     => ( aa_set_nat_o(ord_less_set_nat(Ya),Z)
       => aa_set_nat_o(ord_less_set_nat(Xc),Z) ) ) ).

% order_le_less_trans
tff(fact_4739_order__le__less__trans,axiom,
    ! [Xc: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Z) ) ) ).

% order_le_less_trans
tff(fact_4740_order__le__less__trans,axiom,
    ! [Xc: num,Ya: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Z) ) ) ).

% order_le_less_trans
tff(fact_4741_order__le__less__trans,axiom,
    ! [Xc: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Z) ) ) ).

% order_le_less_trans
tff(fact_4742_order__le__less__trans,axiom,
    ! [Xc: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Z) ) ) ).

% order_le_less_trans
tff(fact_4743_order__less__le__trans,axiom,
    ! [Xc: real,Ya: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Z)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Z) ) ) ).

% order_less_le_trans
tff(fact_4744_order__less__le__trans,axiom,
    ! [Xc: set_nat,Ya: set_nat,Z: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(Xc),Ya)
     => ( aa_set_nat_o(ord_less_eq_set_nat(Ya),Z)
       => aa_set_nat_o(ord_less_set_nat(Xc),Z) ) ) ).

% order_less_le_trans
tff(fact_4745_order__less__le__trans,axiom,
    ! [Xc: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Z)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Z) ) ) ).

% order_less_le_trans
tff(fact_4746_order__less__le__trans,axiom,
    ! [Xc: num,Ya: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Z)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Z) ) ) ).

% order_less_le_trans
tff(fact_4747_order__less__le__trans,axiom,
    ! [Xc: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Z) ) ) ).

% order_less_le_trans
tff(fact_4748_order__less__le__trans,axiom,
    ! [Xc: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Z) ) ) ).

% order_less_le_trans
tff(fact_4749_order__le__less__subst1,axiom,
    ! [A2: real,F2: fun_real_real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(F2,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(F2,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_4750_order__le__less__subst1,axiom,
    ! [A2: real,F2: fun_rat_real,B2: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_rat_real(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_rat_real(F2,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_4751_order__le__less__subst1,axiom,
    ! [A2: real,F2: fun_num_real,B2: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_num_real(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F2,X3)),aa_num_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_num_real(F2,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_4752_order__le__less__subst1,axiom,
    ! [A2: real,F2: fun_nat_real,B2: nat,C2: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_nat_real(F2,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F2,X3)),aa_nat_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_nat_real(F2,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_4753_order__le__less__subst1,axiom,
    ! [A2: real,F2: fun_int_real,B2: int,C2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_int_real(F2,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F2,X3)),aa_int_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_int_real(F2,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_4754_order__le__less__subst1,axiom,
    ! [A2: rat,F2: fun_real_rat,B2: real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_real_rat(F2,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_real_rat(F2,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_4755_order__le__less__subst1,axiom,
    ! [A2: rat,F2: fun_rat_rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(F2,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_4756_order__le__less__subst1,axiom,
    ! [A2: rat,F2: fun_num_rat,B2: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_num_rat(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_num_rat(F2,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_4757_order__le__less__subst1,axiom,
    ! [A2: rat,F2: fun_nat_rat,B2: nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_nat_rat(F2,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_nat_rat(F2,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_4758_order__le__less__subst1,axiom,
    ! [A2: rat,F2: fun_int_rat,B2: int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_int_rat(F2,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F2,X3)),aa_int_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_int_rat(F2,C2)) ) ) ) ).

% order_le_less_subst1
tff(fact_4759_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,A2)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_4760_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,A2)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_4761_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F2,A2)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_4762_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_nat,C2: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F2,A2)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_4763_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_int,C2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F2,A2)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_4764_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F2: fun_num_real,C2: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F2,B2)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F2,X3)),aa_num_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F2,A2)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_4765_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F2: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F2,B2)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F2,A2)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_4766_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F2: fun_num_num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F2,B2)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F2,A2)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_4767_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F2: fun_num_nat,C2: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F2,B2)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F2,A2)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_4768_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F2: fun_num_int,C2: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F2,B2)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F2,X3)),aa_num_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F2,A2)),C2) ) ) ) ).

% order_le_less_subst2
tff(fact_4769_order__less__le__subst1,axiom,
    ! [A2: real,F2: fun_rat_real,B2: rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_rat_real(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_rat_real(F2,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_4770_order__less__le__subst1,axiom,
    ! [A2: rat,F2: fun_rat_rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(F2,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_4771_order__less__le__subst1,axiom,
    ! [A2: num,F2: fun_rat_num,B2: rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),aa_rat_num(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F2,X3)),aa_rat_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A2),aa_rat_num(F2,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_4772_order__less__le__subst1,axiom,
    ! [A2: nat,F2: fun_rat_nat,B2: rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_rat_nat(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F2,X3)),aa_rat_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_rat_nat(F2,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_4773_order__less__le__subst1,axiom,
    ! [A2: int,F2: fun_rat_int,B2: rat,C2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_rat_int(F2,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F2,X3)),aa_rat_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_rat_int(F2,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_4774_order__less__le__subst1,axiom,
    ! [A2: real,F2: fun_num_real,B2: num,C2: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_num_real(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F2,X3)),aa_num_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_num_real(F2,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_4775_order__less__le__subst1,axiom,
    ! [A2: rat,F2: fun_num_rat,B2: num,C2: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_num_rat(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_num_rat(F2,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_4776_order__less__le__subst1,axiom,
    ! [A2: num,F2: fun_num_num,B2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),aa_num_num(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F2,X3)),aa_num_num(F2,Y)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_num,A2),aa_num_num(F2,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_4777_order__less__le__subst1,axiom,
    ! [A2: nat,F2: fun_num_nat,B2: num,C2: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_num_nat(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F2,X3)),aa_num_nat(F2,Y)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),aa_num_nat(F2,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_4778_order__less__le__subst1,axiom,
    ! [A2: int,F2: fun_num_int,B2: num,C2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_num_int(F2,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F2,X3)),aa_num_int(F2,Y)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_num_int(F2,C2)) ) ) ) ).

% order_less_le_subst1
tff(fact_4779_order__less__le__subst2,axiom,
    ! [A2: real,B2: real,F2: fun_real_real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F2,B2)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,X3)),aa_real_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,A2)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_4780_order__less__le__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_real,C2: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,X3)),aa_rat_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F2,A2)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_4781_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F2: fun_num_real,C2: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F2,B2)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F2,X3)),aa_num_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F2,A2)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_4782_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F2: fun_nat_real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F2,B2)),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F2,X3)),aa_nat_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F2,A2)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_4783_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F2: fun_int_real,C2: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F2,B2)),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F2,X3)),aa_int_real(F2,Y)) )
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(F2,A2)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_4784_order__less__le__subst2,axiom,
    ! [A2: real,B2: real,F2: fun_real_rat,C2: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F2,B2)),C2)
       => ( ! [X3: real,Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F2,X3)),aa_real_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F2,A2)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_4785_order__less__le__subst2,axiom,
    ! [A2: rat,B2: rat,F2: fun_rat_rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F2,B2)),C2)
       => ( ! [X3: rat,Y: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,X3)),aa_rat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F2,A2)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_4786_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F2: fun_num_rat,C2: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F2,B2)),C2)
       => ( ! [X3: num,Y: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F2,X3)),aa_num_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F2,A2)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_4787_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F2: fun_nat_rat,C2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,B2)),C2)
       => ( ! [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F2,X3)),aa_nat_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F2,A2)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_4788_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F2: fun_int_rat,C2: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F2,B2)),C2)
       => ( ! [X3: int,Y: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F2,X3)),aa_int_rat(F2,Y)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F2,A2)),C2) ) ) ) ).

% order_less_le_subst2
tff(fact_4789_linorder__le__less__linear,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ).

% linorder_le_less_linear
tff(fact_4790_linorder__le__less__linear,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Xc) ) ).

% linorder_le_less_linear
tff(fact_4791_linorder__le__less__linear,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Xc) ) ).

% linorder_le_less_linear
tff(fact_4792_linorder__le__less__linear,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Xc) ) ).

% linorder_le_less_linear
tff(fact_4793_linorder__le__less__linear,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Xc) ) ).

% linorder_le_less_linear
tff(fact_4794_order__le__imp__less__or__eq,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4795_order__le__imp__less__or__eq,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
     => ( aa_set_nat_o(ord_less_set_nat(Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4796_order__le__imp__less__or__eq,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4797_order__le__imp__less__or__eq,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4798_order__le__imp__less__or__eq,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4799_order__le__imp__less__or__eq,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4800_verit__sum__simplify,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(A2),zero_zero_complex) = A2 ).

% verit_sum_simplify
tff(fact_4801_verit__sum__simplify,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),zero_zero_real) = A2 ).

% verit_sum_simplify
tff(fact_4802_verit__sum__simplify,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(A2),zero_zero_rat) = A2 ).

% verit_sum_simplify
tff(fact_4803_verit__sum__simplify,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),zero_zero_nat) = A2 ).

% verit_sum_simplify
tff(fact_4804_verit__sum__simplify,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(A2),zero_zero_int) = A2 ).

% verit_sum_simplify
tff(fact_4805_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one != aa_num_num(bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_4806_bot_Oextremum,axiom,
    ! [A2: set_real] : aa_set_real_o(ord_less_eq_set_real(bot_bot_set_real),A2) ).

% bot.extremum
tff(fact_4807_bot_Oextremum,axiom,
    ! [A2: set_o] : aa_set_o_o(ord_less_eq_set_o(bot_bot_set_o),A2) ).

% bot.extremum
tff(fact_4808_bot_Oextremum,axiom,
    ! [A2: set_int] : aa_set_int_o(ord_less_eq_set_int(bot_bot_set_int),A2) ).

% bot.extremum
tff(fact_4809_bot_Oextremum,axiom,
    ! [A2: set_nat] : aa_set_nat_o(ord_less_eq_set_nat(bot_bot_set_nat),A2) ).

% bot.extremum
tff(fact_4810_bot_Oextremum,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,bot_bot_nat),A2) ).

% bot.extremum
tff(fact_4811_bot_Oextremum__unique,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),bot_bot_set_real)
    <=> ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_4812_bot_Oextremum__unique,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),bot_bot_set_o)
    <=> ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_unique
tff(fact_4813_bot_Oextremum__unique,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),bot_bot_set_int)
    <=> ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_4814_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),bot_bot_set_nat)
    <=> ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_4815_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),bot_bot_nat)
    <=> ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
tff(fact_4816_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A2),bot_bot_set_real)
     => ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_4817_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A2),bot_bot_set_o)
     => ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
tff(fact_4818_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A2),bot_bot_set_int)
     => ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_4819_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A2),bot_bot_set_nat)
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_4820_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),bot_bot_nat)
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_4821_bot_Oextremum__strict,axiom,
    ! [A2: set_real] : ~ aa_set_real_o(ord_less_set_real(A2),bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_4822_bot_Oextremum__strict,axiom,
    ! [A2: set_o] : ~ aa_set_o_o(ord_less_set_o(A2),bot_bot_set_o) ).

% bot.extremum_strict
tff(fact_4823_bot_Oextremum__strict,axiom,
    ! [A2: set_nat] : ~ aa_set_nat_o(ord_less_set_nat(A2),bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_4824_bot_Oextremum__strict,axiom,
    ! [A2: set_int] : ~ aa_set_int_o(ord_less_set_int(A2),bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_4825_bot_Oextremum__strict,axiom,
    ! [A2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),bot_bot_nat) ).

% bot.extremum_strict
tff(fact_4826_bot_Onot__eq__extremum,axiom,
    ! [A2: set_real] :
      ( ( A2 != bot_bot_set_real )
    <=> aa_set_real_o(ord_less_set_real(bot_bot_set_real),A2) ) ).

% bot.not_eq_extremum
tff(fact_4827_bot_Onot__eq__extremum,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
    <=> aa_set_o_o(ord_less_set_o(bot_bot_set_o),A2) ) ).

% bot.not_eq_extremum
tff(fact_4828_bot_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_set_nat(bot_bot_set_nat),A2) ) ).

% bot.not_eq_extremum
tff(fact_4829_bot_Onot__eq__extremum,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
    <=> aa_set_int_o(ord_less_set_int(bot_bot_set_int),A2) ) ).

% bot.not_eq_extremum
tff(fact_4830_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,bot_bot_nat),A2) ) ).

% bot.not_eq_extremum
tff(fact_4831_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X33: num] : aa_num_num(bit0,X22) != aa_num_num(bit1,X33) ).

% verit_eq_simplify(14)
tff(fact_4832_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_13: nat] : aa_nat_o(P,X_13)
       => ? [N: nat] :
            ( ~ aa_nat_o(P,N)
            & aa_nat_o(P,aa_nat_nat(suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_4833_real__arch__simple,axiom,
    ! [Xc: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% real_arch_simple
tff(fact_4834_real__arch__simple,axiom,
    ! [Xc: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% real_arch_simple
tff(fact_4835_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] : one != aa_num_num(bit1,X33) ).

% verit_eq_simplify(12)
tff(fact_4836_reals__Archimedean2,axiom,
    ! [Xc: rat] :
    ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),aa_nat_rat(semiri681578069525770553at_rat,N)) ).

% reals_Archimedean2
tff(fact_4837_reals__Archimedean2,axiom,
    ! [Xc: real] :
    ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_nat_real(semiri5074537144036343181t_real,N)) ).

% reals_Archimedean2
tff(fact_4838_max__absorb2,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
     => ( ord_max_set_nat(Xc,Ya) = Ya ) ) ).

% max_absorb2
tff(fact_4839_max__absorb2,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
     => ( aa_rat_rat(ord_max_rat(Xc),Ya) = Ya ) ) ).

% max_absorb2
tff(fact_4840_max__absorb2,axiom,
    ! [Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xc),Ya)
     => ( aa_num_num(ord_max_num(Xc),Ya) = Ya ) ) ).

% max_absorb2
tff(fact_4841_max__absorb2,axiom,
    ! [Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Ya)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),Ya) = Ya ) ) ).

% max_absorb2
tff(fact_4842_max__absorb2,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
     => ( aa_int_int(ord_max_int(Xc),Ya) = Ya ) ) ).

% max_absorb2
tff(fact_4843_max__absorb1,axiom,
    ! [Ya: set_nat,Xc: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Ya),Xc)
     => ( ord_max_set_nat(Xc,Ya) = Xc ) ) ).

% max_absorb1
tff(fact_4844_max__absorb1,axiom,
    ! [Ya: rat,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc)
     => ( aa_rat_rat(ord_max_rat(Xc),Ya) = Xc ) ) ).

% max_absorb1
tff(fact_4845_max__absorb1,axiom,
    ! [Ya: num,Xc: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ya),Xc)
     => ( aa_num_num(ord_max_num(Xc),Ya) = Xc ) ) ).

% max_absorb1
tff(fact_4846_max__absorb1,axiom,
    ! [Ya: nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),Ya) = Xc ) ) ).

% max_absorb1
tff(fact_4847_max__absorb1,axiom,
    ! [Ya: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc)
     => ( aa_int_int(ord_max_int(Xc),Ya) = Xc ) ) ).

% max_absorb1
tff(fact_4848_max__def,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ord_max_set_nat(A2,B2) = $ite(aa_set_nat_o(ord_less_eq_set_nat(A2),B2),B2,A2) ).

% max_def
tff(fact_4849_max__def,axiom,
    ! [A2: rat,B2: rat] :
      aa_rat_rat(ord_max_rat(A2),B2) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2),B2,A2) ).

% max_def
tff(fact_4850_max__def,axiom,
    ! [A2: num,B2: num] :
      aa_num_num(ord_max_num(A2),B2) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2),B2,A2) ).

% max_def
tff(fact_4851_max__def,axiom,
    ! [A2: nat,B2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2),B2,A2) ).

% max_def
tff(fact_4852_max__def,axiom,
    ! [A2: int,B2: int] :
      aa_int_int(ord_max_int(A2),B2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2),B2,A2) ).

% max_def
tff(fact_4853_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_o_VEBT_VEBT(vEBT_Leaf($false),$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_4854_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv2: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_o_VEBT_VEBT(vEBT_Leaf($true),(Uv2))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_4855_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_o_VEBT_VEBT(vEBT_Leaf((Uu)),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_4856_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw)),Xc) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_4857_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2),Uz2)),Xc) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_4858_int__ops_I3_J,axiom,
    ! [Nb: num] : aa_nat_int(semiri1314217659103216013at_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% int_ops(3)
tff(fact_4859_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_4860_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ) ).

% nat_int_comparison(2)
tff(fact_4861_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ) ).

% nat_int_comparison(3)
tff(fact_4862_int__ops_I5_J,axiom,
    ! [A2: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% int_ops(5)
tff(fact_4863_int__plus,axiom,
    ! [Nb: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% int_plus
tff(fact_4864_int__ops_I7_J,axiom,
    ! [A2: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)) = aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% int_ops(7)
tff(fact_4865_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Vaa: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Vaa,Vb2),Vc2)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_4866_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_4867_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),Xc) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_4868_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,Xc: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw)),Xc) ).

% vebt_member.simps(2)
tff(fact_4869_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xc: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),Xc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
        $ite(Xc = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_4870_ex__less__of__nat__mult,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xc)
     => ? [N: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),aa_rat_rat(times_times_rat(aa_nat_rat(semiri681578069525770553at_rat,N)),Xc)) ) ).

% ex_less_of_nat_mult
tff(fact_4871_ex__less__of__nat__mult,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ? [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),Xc)) ) ).

% ex_less_of_nat_mult
tff(fact_4872_int__ops_I4_J,axiom,
    ! [A2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A2)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A2)),one_one_int) ).

% int_ops(4)
tff(fact_4873_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_4874_less__1__helper,axiom,
    ! [Nb: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),M)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(Nb),one_one_int)),M) ) ).

% less_1_helper
tff(fact_4875_vebt__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),Xc)
    <=> $ite(
          Xc = zero_zero_nat,
          (A2),
          $ite(Xc = one_one_nat,(B2),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_4876_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Xc: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2),Uz2)),Xc) ).

% vebt_member.simps(3)
tff(fact_4877_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xc: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xc)
     => ( ! [Uv: $o] : Xc != aa_o_VEBT_VEBT(vEBT_Leaf($true),(Uv))
       => ( ! [Uu2: $o] : Xc != aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),$true)
         => ~ ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_4878_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xc)
     => ( ( Xc != aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) )
       => ~ ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux),Uy) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_4879_member__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r(Ta),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,one)))))) ) ).

% member_bound_height
tff(fact_4880_max_Oabsorb3,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_4881_max_Oabsorb3,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
     => ( aa_rat_rat(ord_max_rat(A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_4882_max_Oabsorb3,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
     => ( aa_num_num(ord_max_num(A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_4883_max_Oabsorb3,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_4884_max_Oabsorb3,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
     => ( aa_int_int(ord_max_int(A2),B2) = A2 ) ) ).

% max.absorb3
tff(fact_4885_max_Oabsorb4,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_4886_max_Oabsorb4,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ( aa_rat_rat(ord_max_rat(A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_4887_max_Oabsorb4,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B2)
     => ( aa_num_num(ord_max_num(A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_4888_max_Oabsorb4,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_4889_max_Oabsorb4,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => ( aa_int_int(ord_max_int(A2),B2) = B2 ) ) ).

% max.absorb4
tff(fact_4890_max__less__iff__conj,axiom,
    ! [Xc: real,Ya: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xc),Ya)),Z)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Z)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Z) ) ) ).

% max_less_iff_conj
tff(fact_4891_max__less__iff__conj,axiom,
    ! [Xc: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(Xc),Ya)),Z)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Z)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ya),Z) ) ) ).

% max_less_iff_conj
tff(fact_4892_max__less__iff__conj,axiom,
    ! [Xc: num,Ya: num,Z: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(ord_max_num(Xc),Ya)),Z)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xc),Z)
        & aa_num_o(aa_num_fun_num_o(ord_less_num,Ya),Z) ) ) ).

% max_less_iff_conj
tff(fact_4893_max__less__iff__conj,axiom,
    ! [Xc: nat,Ya: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),Ya)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Z) ) ) ).

% max_less_iff_conj
tff(fact_4894_max__less__iff__conj,axiom,
    ! [Xc: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(Xc),Ya)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Z) ) ) ).

% max_less_iff_conj
tff(fact_4895_max_Obounded__iff,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B2),C2)),A2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A2) ) ) ).

% max.bounded_iff
tff(fact_4896_max_Obounded__iff,axiom,
    ! [B2: num,C2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B2),C2)),A2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A2) ) ) ).

% max.bounded_iff
tff(fact_4897_max_Obounded__iff,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)),A2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A2) ) ) ).

% max.bounded_iff
tff(fact_4898_max_Obounded__iff,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B2),C2)),A2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A2) ) ) ).

% max.bounded_iff
tff(fact_4899_max_Oabsorb2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_rat(ord_max_rat(A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_4900_max_Oabsorb2,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
     => ( aa_num_num(ord_max_num(A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_4901_max_Oabsorb2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_4902_max_Oabsorb2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_int(ord_max_int(A2),B2) = B2 ) ) ).

% max.absorb2
tff(fact_4903_max_Oabsorb1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_rat(ord_max_rat(A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_4904_max_Oabsorb1,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
     => ( aa_num_num(ord_max_num(A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_4905_max_Oabsorb1,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_4906_max_Oabsorb1,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_int(ord_max_int(A2),B2) = A2 ) ) ).

% max.absorb1
tff(fact_4907_enat__ord__number_I1_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(1)
tff(fact_4908_enat__ord__number_I2_J,axiom,
    ! [M: num,Nb: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(M)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(2)
tff(fact_4909_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Ya: extended_enat,Xc: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Z),Ya)
     => ( plus_p3455044024723400733d_enat(Xc,minus_3235023915231533773d_enat(Ya,Z)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xc,Ya),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_4910_max_Omono,axiom,
    ! [C2: rat,A2: rat,D2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,D2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(C2),D2)),aa_rat_rat(ord_max_rat(A2),B2)) ) ) ).

% max.mono
tff(fact_4911_max_Omono,axiom,
    ! [C2: num,A2: num,D2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,D2),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(C2),D2)),aa_num_num(ord_max_num(A2),B2)) ) ) ).

% max.mono
tff(fact_4912_max_Omono,axiom,
    ! [C2: nat,A2: nat,D2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C2),D2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ) ) ).

% max.mono
tff(fact_4913_max_Omono,axiom,
    ! [C2: int,A2: int,D2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(C2),D2)),aa_int_int(ord_max_int(A2),B2)) ) ) ).

% max.mono
tff(fact_4914_max_OorderE,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( A2 = aa_rat_rat(ord_max_rat(A2),B2) ) ) ).

% max.orderE
tff(fact_4915_max_OorderE,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
     => ( A2 = aa_num_num(ord_max_num(A2),B2) ) ) ).

% max.orderE
tff(fact_4916_max_OorderE,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) ) ) ).

% max.orderE
tff(fact_4917_max_OorderE,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( A2 = aa_int_int(ord_max_int(A2),B2) ) ) ).

% max.orderE
tff(fact_4918_max_OorderI,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(ord_max_rat(A2),B2) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2) ) ).

% max.orderI
tff(fact_4919_max_OorderI,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 = aa_num_num(ord_max_num(A2),B2) )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2) ) ).

% max.orderI
tff(fact_4920_max_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2) ) ).

% max.orderI
tff(fact_4921_max_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(ord_max_int(A2),B2) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2) ) ).

% max.orderI
tff(fact_4922_max_OboundedE,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B2),C2)),A2)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A2) ) ) ).

% max.boundedE
tff(fact_4923_max_OboundedE,axiom,
    ! [B2: num,C2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B2),C2)),A2)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A2) ) ) ).

% max.boundedE
tff(fact_4924_max_OboundedE,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)),A2)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A2) ) ) ).

% max.boundedE
tff(fact_4925_max_OboundedE,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B2),C2)),A2)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A2) ) ) ).

% max.boundedE
tff(fact_4926_max_OboundedI,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(ord_max_rat(B2),C2)),A2) ) ) ).

% max.boundedI
tff(fact_4927_max_OboundedI,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(ord_max_num(B2),C2)),A2) ) ) ).

% max.boundedI
tff(fact_4928_max_OboundedI,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)),A2) ) ) ).

% max.boundedI
tff(fact_4929_max_OboundedI,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ord_max_int(B2),C2)),A2) ) ) ).

% max.boundedI
tff(fact_4930_max_Oorder__iff,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> ( A2 = aa_rat_rat(ord_max_rat(A2),B2) ) ) ).

% max.order_iff
tff(fact_4931_max_Oorder__iff,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> ( A2 = aa_num_num(ord_max_num(A2),B2) ) ) ).

% max.order_iff
tff(fact_4932_max_Oorder__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) ) ) ).

% max.order_iff
tff(fact_4933_max_Oorder__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> ( A2 = aa_int_int(ord_max_int(A2),B2) ) ) ).

% max.order_iff
tff(fact_4934_max_Ocobounded1,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(ord_max_rat(A2),B2)) ).

% max.cobounded1
tff(fact_4935_max_Ocobounded1,axiom,
    ! [A2: num,B2: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),aa_num_num(ord_max_num(A2),B2)) ).

% max.cobounded1
tff(fact_4936_max_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ).

% max.cobounded1
tff(fact_4937_max_Ocobounded1,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(ord_max_int(A2),B2)) ).

% max.cobounded1
tff(fact_4938_max_Ocobounded2,axiom,
    ! [B2: rat,A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(ord_max_rat(A2),B2)) ).

% max.cobounded2
tff(fact_4939_max_Ocobounded2,axiom,
    ! [B2: num,A2: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),aa_num_num(ord_max_num(A2),B2)) ).

% max.cobounded2
tff(fact_4940_max_Ocobounded2,axiom,
    ! [B2: nat,A2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ).

% max.cobounded2
tff(fact_4941_max_Ocobounded2,axiom,
    ! [B2: int,A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(ord_max_int(A2),B2)) ).

% max.cobounded2
tff(fact_4942_le__max__iff__disj,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),aa_rat_rat(ord_max_rat(Xc),Ya))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Xc)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Z),Ya) ) ) ).

% le_max_iff_disj
tff(fact_4943_le__max__iff__disj,axiom,
    ! [Z: num,Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),aa_num_num(ord_max_num(Xc),Ya))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Xc)
        | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Z),Ya) ) ) ).

% le_max_iff_disj
tff(fact_4944_le__max__iff__disj,axiom,
    ! [Z: nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),Ya))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Xc)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Ya) ) ) ).

% le_max_iff_disj
tff(fact_4945_le__max__iff__disj,axiom,
    ! [Z: int,Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_int_int(ord_max_int(Xc),Ya))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Xc)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Ya) ) ) ).

% le_max_iff_disj
tff(fact_4946_max_Oabsorb__iff1,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A2)
    <=> ( aa_rat_rat(ord_max_rat(A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_4947_max_Oabsorb__iff1,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A2)
    <=> ( aa_num_num(ord_max_num(A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_4948_max_Oabsorb__iff1,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_4949_max_Oabsorb__iff1,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A2)
    <=> ( aa_int_int(ord_max_int(A2),B2) = A2 ) ) ).

% max.absorb_iff1
tff(fact_4950_max_Oabsorb__iff2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
    <=> ( aa_rat_rat(ord_max_rat(A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_4951_max_Oabsorb__iff2,axiom,
    ! [A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),B2)
    <=> ( aa_num_num(ord_max_num(A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_4952_max_Oabsorb__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_4953_max_Oabsorb__iff2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
    <=> ( aa_int_int(ord_max_int(A2),B2) = B2 ) ) ).

% max.absorb_iff2
tff(fact_4954_max_OcoboundedI1,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(ord_max_rat(A2),B2)) ) ).

% max.coboundedI1
tff(fact_4955_max_OcoboundedI1,axiom,
    ! [C2: num,A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),A2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),aa_num_num(ord_max_num(A2),B2)) ) ).

% max.coboundedI1
tff(fact_4956_max_OcoboundedI1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ) ).

% max.coboundedI1
tff(fact_4957_max_OcoboundedI1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(ord_max_int(A2),B2)) ) ).

% max.coboundedI1
tff(fact_4958_max_OcoboundedI2,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C2),aa_rat_rat(ord_max_rat(A2),B2)) ) ).

% max.coboundedI2
tff(fact_4959_max_OcoboundedI2,axiom,
    ! [C2: num,B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),B2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C2),aa_num_num(ord_max_num(A2),B2)) ) ).

% max.coboundedI2
tff(fact_4960_max_OcoboundedI2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ) ).

% max.coboundedI2
tff(fact_4961_max_OcoboundedI2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C2),aa_int_int(ord_max_int(A2),B2)) ) ).

% max.coboundedI2
tff(fact_4962_less__max__iff__disj,axiom,
    ! [Z: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,Xc),Ya))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Xc)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Ya) ) ) ).

% less_max_iff_disj
tff(fact_4963_less__max__iff__disj,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),aa_rat_rat(ord_max_rat(Xc),Ya))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Xc)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Ya) ) ) ).

% less_max_iff_disj
tff(fact_4964_less__max__iff__disj,axiom,
    ! [Z: num,Xc: num,Ya: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),aa_num_num(ord_max_num(Xc),Ya))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Xc)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Z),Ya) ) ) ).

% less_max_iff_disj
tff(fact_4965_less__max__iff__disj,axiom,
    ! [Z: nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xc),Ya))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xc)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Ya) ) ) ).

% less_max_iff_disj
tff(fact_4966_less__max__iff__disj,axiom,
    ! [Z: int,Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(ord_max_int(Xc),Ya))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Xc)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Ya) ) ) ).

% less_max_iff_disj
tff(fact_4967_max_Ostrict__boundedE,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,B2),C2)),A2)
     => ~ ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
         => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A2) ) ) ).

% max.strict_boundedE
tff(fact_4968_max_Ostrict__boundedE,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(ord_max_rat(B2),C2)),A2)
     => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A2) ) ) ).

% max.strict_boundedE
tff(fact_4969_max_Ostrict__boundedE,axiom,
    ! [B2: num,C2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(ord_max_num(B2),C2)),A2)
     => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A2) ) ) ).

% max.strict_boundedE
tff(fact_4970_max_Ostrict__boundedE,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B2),C2)),A2)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A2) ) ) ).

% max.strict_boundedE
tff(fact_4971_max_Ostrict__boundedE,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ord_max_int(B2),C2)),A2)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A2) ) ) ).

% max.strict_boundedE
tff(fact_4972_max_Ostrict__order__iff,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
    <=> ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4973_max_Ostrict__order__iff,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A2)
    <=> ( ( A2 = aa_rat_rat(ord_max_rat(A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4974_max_Ostrict__order__iff,axiom,
    ! [B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A2)
    <=> ( ( A2 = aa_num_num(ord_max_num(A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4975_max_Ostrict__order__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A2)
    <=> ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4976_max_Ostrict__order__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A2)
    <=> ( ( A2 = aa_int_int(ord_max_int(A2),B2) )
        & ( A2 != B2 ) ) ) ).

% max.strict_order_iff
tff(fact_4977_max_Ostrict__coboundedI1,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),A2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B2)) ) ).

% max.strict_coboundedI1
tff(fact_4978_max_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),A2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(ord_max_rat(A2),B2)) ) ).

% max.strict_coboundedI1
tff(fact_4979_max_Ostrict__coboundedI1,axiom,
    ! [C2: num,A2: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),A2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),aa_num_num(ord_max_num(A2),B2)) ) ).

% max.strict_coboundedI1
tff(fact_4980_max_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ) ).

% max.strict_coboundedI1
tff(fact_4981_max_Ostrict__coboundedI1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(ord_max_int(A2),B2)) ) ).

% max.strict_coboundedI1
tff(fact_4982_max_Ostrict__coboundedI2,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,C2),aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,A2),B2)) ) ).

% max.strict_coboundedI2
tff(fact_4983_max_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),aa_rat_rat(ord_max_rat(A2),B2)) ) ).

% max.strict_coboundedI2
tff(fact_4984_max_Ostrict__coboundedI2,axiom,
    ! [C2: num,B2: num,A2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,C2),B2)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,C2),aa_num_num(ord_max_num(A2),B2)) ) ).

% max.strict_coboundedI2
tff(fact_4985_max_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A2),B2)) ) ).

% max.strict_coboundedI2
tff(fact_4986_max_Ostrict__coboundedI2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C2),aa_int_int(ord_max_int(A2),B2)) ) ).

% max.strict_coboundedI2
tff(fact_4987_heigt__uplog__rel,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% heigt_uplog_rel
tff(fact_4988_set__bit__0,axiom,
    ! [A2: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A2) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4989_set__bit__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_4990_i0__less,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb)
    <=> ( Nb != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_4991_idiff__0__right,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,zero_z5237406670263579293d_enat) = Nb ).

% idiff_0_right
tff(fact_4992_idiff__0,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Nb) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_4993_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se7879613467334960850it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_4994_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se7879613467334960850it_int(Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% set_bit_negative_int_iff
tff(fact_4995_ceiling__zero,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,zero_zero_rat) = zero_zero_int ).

% ceiling_zero
tff(fact_4996_ceiling__zero,axiom,
    aa_real_int(archim7802044766580827645g_real,zero_zero_real) = zero_zero_int ).

% ceiling_zero
tff(fact_4997_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_4998_ceiling__numeral,axiom,
    ! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_4999_ceiling__le__zero,axiom,
    ! [Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xc)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_5000_ceiling__le__zero,axiom,
    ! [Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_5001_ceiling__le__numeral,axiom,
    ! [Xc: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xc)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_num_real(numeral_numeral_real,V)) ) ).

% ceiling_le_numeral
tff(fact_5002_ceiling__le__numeral,axiom,
    ! [Xc: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_num_rat(numeral_numeral_rat,V)) ) ).

% ceiling_le_numeral
tff(fact_5003_zero__less__ceiling,axiom,
    ! [Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xc))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xc) ) ).

% zero_less_ceiling
tff(fact_5004_zero__less__ceiling,axiom,
    ! [Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc) ) ).

% zero_less_ceiling
tff(fact_5005_numeral__less__ceiling,axiom,
    ! [V: num,Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,V)),Xc) ) ).

% numeral_less_ceiling
tff(fact_5006_numeral__less__ceiling,axiom,
    ! [V: num,Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,Xc))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),Xc) ) ).

% numeral_less_ceiling
tff(fact_5007_ceiling__less__one,axiom,
    ! [Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xc)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_5008_ceiling__less__one,axiom,
    ! [Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_5009_one__le__ceiling,axiom,
    ! [Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xc))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xc) ) ).

% one_le_ceiling
tff(fact_5010_one__le__ceiling,axiom,
    ! [Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc) ) ).

% one_le_ceiling
tff(fact_5011_ceiling__add__numeral,axiom,
    ! [Xc: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_num_real(numeral_numeral_real,V))) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,Xc)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_5012_ceiling__add__numeral,axiom,
    ! [Xc: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xc),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(plus_plus_int(aa_rat_int(archim2889992004027027881ng_rat,Xc)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_5013_ceiling__le__one,axiom,
    ! [Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xc)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real) ) ).

% ceiling_le_one
tff(fact_5014_ceiling__le__one,axiom,
    ! [Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),one_one_rat) ) ).

% ceiling_le_one
tff(fact_5015_one__less__ceiling,axiom,
    ! [Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xc))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xc) ) ).

% one_less_ceiling
tff(fact_5016_one__less__ceiling,axiom,
    ! [Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc) ) ).

% one_less_ceiling
tff(fact_5017_ceiling__add__one,axiom,
    ! [Xc: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xc),one_one_rat)) = aa_int_int(plus_plus_int(aa_rat_int(archim2889992004027027881ng_rat,Xc)),one_one_int) ).

% ceiling_add_one
tff(fact_5018_ceiling__add__one,axiom,
    ! [Xc: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),one_one_real)) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,Xc)),one_one_int) ).

% ceiling_add_one
tff(fact_5019_ceiling__diff__numeral,axiom,
    ! [Xc: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),aa_num_real(numeral_numeral_real,V))) = aa_int_int(minus_minus_int(aa_real_int(archim7802044766580827645g_real,Xc)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_5020_ceiling__diff__numeral,axiom,
    ! [Xc: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(Xc),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(minus_minus_int(aa_rat_int(archim2889992004027027881ng_rat,Xc)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_5021_ceiling__numeral__power,axiom,
    ! [Xc: num,Nb: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xc)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) ).

% ceiling_numeral_power
tff(fact_5022_ceiling__numeral__power,axiom,
    ! [Xc: num,Nb: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xc)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) ).

% ceiling_numeral_power
tff(fact_5023_ceiling__diff__one,axiom,
    ! [Xc: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(Xc),one_one_rat)) = aa_int_int(minus_minus_int(aa_rat_int(archim2889992004027027881ng_rat,Xc)),one_one_int) ).

% ceiling_diff_one
tff(fact_5024_ceiling__diff__one,axiom,
    ! [Xc: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),one_one_real)) = aa_int_int(minus_minus_int(aa_real_int(archim7802044766580827645g_real,Xc)),one_one_int) ).

% ceiling_diff_one
tff(fact_5025_ceiling__less__numeral,axiom,
    ! [Xc: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xc)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_5026_ceiling__less__numeral,axiom,
    ! [Xc: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_5027_numeral__le__ceiling,axiom,
    ! [V: num,Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),Xc) ) ).

% numeral_le_ceiling
tff(fact_5028_numeral__le__ceiling,axiom,
    ! [V: num,Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,Xc))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),Xc) ) ).

% numeral_le_ceiling
tff(fact_5029_not__iless0,axiom,
    ! [Nb: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_5030_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Nb: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M2),N)
             => aa_Extended_enat_o(P,M2) )
         => aa_Extended_enat_o(P,N) )
     => aa_Extended_enat_o(P,Nb) ) ).

% enat_less_induct
tff(fact_5031_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(M,Nb))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),M)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_5032_set__bit__greater__eq,axiom,
    ! [K: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se7879613467334960850it_int(Nb),K)) ).

% set_bit_greater_eq
tff(fact_5033_ceiling__mono,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Ya)),aa_real_int(archim7802044766580827645g_real,Xc)) ) ).

% ceiling_mono
tff(fact_5034_ceiling__mono,axiom,
    ! [Ya: rat,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ya),Xc)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Ya)),aa_rat_int(archim2889992004027027881ng_rat,Xc)) ) ).

% ceiling_mono
tff(fact_5035_ceiling__less__cancel,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),aa_rat_int(archim2889992004027027881ng_rat,Ya))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya) ) ).

% ceiling_less_cancel
tff(fact_5036_ceiling__less__cancel,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xc)),aa_real_int(archim7802044766580827645g_real,Ya))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ).

% ceiling_less_cancel
tff(fact_5037_ceiling__add__le,axiom,
    ! [Xc: rat,Ya: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xc),Ya))),aa_int_int(plus_plus_int(aa_rat_int(archim2889992004027027881ng_rat,Xc)),aa_rat_int(archim2889992004027027881ng_rat,Ya))) ).

% ceiling_add_le
tff(fact_5038_ceiling__add__le,axiom,
    ! [Xc: real,Ya: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya))),aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,Xc)),aa_real_int(archim7802044766580827645g_real,Ya))) ).

% ceiling_add_le
tff(fact_5039_mult__ceiling__le,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2))),aa_int_int(times_times_int(aa_real_int(archim7802044766580827645g_real,A2)),aa_real_int(archim7802044766580827645g_real,B2))) ) ) ).

% mult_ceiling_le
tff(fact_5040_mult__ceiling__le,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(times_times_rat(A2),B2))),aa_int_int(times_times_int(aa_rat_int(archim2889992004027027881ng_rat,A2)),aa_rat_int(archim2889992004027027881ng_rat,B2))) ) ) ).

% mult_ceiling_le
tff(fact_5041_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] : aa_rat_rat(plus_plus_rat(aa_nat_rat(semiri681578069525770553at_rat,A2)),aa_nat_rat(semiri681578069525770553at_rat,B2)) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ).

% Abs_fnat_hom_add
tff(fact_5042_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,A2)),aa_nat_real(semiri5074537144036343181t_real,B2)) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ).

% Abs_fnat_hom_add
tff(fact_5043_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] : aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ).

% Abs_fnat_hom_add
tff(fact_5044_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,A2)),aa_nat_nat(semiri1316708129612266289at_nat,B2)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ).

% Abs_fnat_hom_add
tff(fact_5045_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] : aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,A2)),aa_nat_complex(semiri8010041392384452111omplex,B2)) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ).

% Abs_fnat_hom_add
tff(fact_5046_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] : plus_p5714425477246183910nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,A2),aa_nat_Code_integer(semiri4939895301339042750nteger,B2)) = aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ).

% Abs_fnat_hom_add
tff(fact_5047_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_5048_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))))),one_one_int) ) ) ).

% ceiling_log2_div2
tff(fact_5049_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_5050_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,K) != zero_zero_rat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_5051_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,K) != zero_zero_real )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_5052_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,K) != zero_zero_int )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_5053_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,K) != zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_5054_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,K) != zero_zero_complex )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_5055_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( aa_nat_Code_integer(semiri4939895301339042750nteger,K) != zero_z3403309356797280102nteger )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K) ) ).

% of_nat_gt_0
tff(fact_5056_log__ceil__idem,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc)
     => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xc)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xc)))) ) ) ).

% log_ceil_idem
tff(fact_5057_succ__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_s_u_c_c(Ta),Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_5058_pred__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_p_r_e_d(Ta),Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_5059_insert__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_i_n_s_e_r_t(Ta),Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_5060_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K6: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),K6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H))),K6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Z),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_5061_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K6: real,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),K6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Z),H))),K6)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Z),H)),Nb)),aa_nat_complex(power_power_complex(Z),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Z),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_5062_succ__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_s_u_c_c(Ta),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))))) ) ).

% succ_bound_height
tff(fact_5063_pred__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d(Ta),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))) ) ).

% pred_bound_height
tff(fact_5064_of__int__ceiling__cancel,axiom,
    ! [Xc: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xc)) = Xc )
    <=> ? [N6: int] : Xc = aa_int_rat(ring_1_of_int_rat,N6) ) ).

% of_int_ceiling_cancel
tff(fact_5065_of__int__ceiling__cancel,axiom,
    ! [Xc: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xc)) = Xc )
    <=> ? [N6: int] : Xc = aa_int_real(ring_1_of_int_real,N6) ) ).

% of_int_ceiling_cancel
tff(fact_5066_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = zero_zero_int )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5067_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = zero_zero_real )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5068_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = zero_zero_rat )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5069_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ring_17405671764205052669omplex(Z) = zero_zero_complex )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
tff(fact_5070_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int = aa_int_int(ring_1_of_int_int,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5071_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real = aa_int_real(ring_1_of_int_real,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5072_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat = aa_int_rat(ring_1_of_int_rat,Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5073_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex = ring_17405671764205052669omplex(Z) )
    <=> ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
tff(fact_5074_of__int__0,axiom,
    aa_int_int(ring_1_of_int_int,zero_zero_int) = zero_zero_int ).

% of_int_0
tff(fact_5075_of__int__0,axiom,
    aa_int_real(ring_1_of_int_real,zero_zero_int) = zero_zero_real ).

% of_int_0
tff(fact_5076_of__int__0,axiom,
    aa_int_rat(ring_1_of_int_rat,zero_zero_int) = zero_zero_rat ).

% of_int_0
tff(fact_5077_of__int__0,axiom,
    ring_17405671764205052669omplex(zero_zero_int) = zero_zero_complex ).

% of_int_0
tff(fact_5078_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( ring_17405671764205052669omplex(Z) = numera6690914467698888265omplex(Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5079_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Z) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5080_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Z) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5081_of__int__eq__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( ( aa_int_int(ring_1_of_int_int,Z) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Z = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_5082_of__int__numeral,axiom,
    ! [K: num] : ring_17405671764205052669omplex(aa_num_int(numeral_numeral_int,K)) = numera6690914467698888265omplex(K) ).

% of_int_numeral
tff(fact_5083_of__int__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_num_int(numeral_numeral_int,K)) = aa_num_real(numeral_numeral_real,K) ).

% of_int_numeral
tff(fact_5084_of__int__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_num_int(numeral_numeral_int,K)) = aa_num_rat(numeral_numeral_rat,K) ).

% of_int_numeral
tff(fact_5085_of__int__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,K) ).

% of_int_numeral
tff(fact_5086_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_5087_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_5088_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ).

% of_int_le_iff
tff(fact_5089_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_5090_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_5091_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ).

% of_int_less_iff
tff(fact_5092_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(plus_plus_int(W),Z)) = aa_int_int(plus_plus_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_add
tff(fact_5093_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_add
tff(fact_5094_of__int__add,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(W),Z)) = aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_add
tff(fact_5095_of__int__add,axiom,
    ! [W: int,Z: int] : ring_17405671764205052669omplex(aa_int_int(plus_plus_int(W),Z)) = aa_complex_complex(plus_plus_complex(ring_17405671764205052669omplex(W)),ring_17405671764205052669omplex(Z)) ).

% of_int_add
tff(fact_5096_of__int__mult,axiom,
    ! [W: int,Z: int] : ring_17405671764205052669omplex(aa_int_int(times_times_int(W),Z)) = aa_complex_complex(times_times_complex(ring_17405671764205052669omplex(W)),ring_17405671764205052669omplex(Z)) ).

% of_int_mult
tff(fact_5097_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(times_times_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_mult
tff(fact_5098_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(times_times_int(W),Z)) = aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_mult
tff(fact_5099_of__int__mult,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(times_times_int(W),Z)) = aa_int_int(times_times_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_mult
tff(fact_5100_of__int__diff,axiom,
    ! [W: int,Z: int] : ring_17405671764205052669omplex(aa_int_int(minus_minus_int(W),Z)) = aa_complex_complex(minus_minus_complex(ring_17405671764205052669omplex(W)),ring_17405671764205052669omplex(Z)) ).

% of_int_diff
tff(fact_5101_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_real(ring_1_of_int_real,aa_int_int(minus_minus_int(W),Z)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Z)) ).

% of_int_diff
tff(fact_5102_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(minus_minus_int(W),Z)) = aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Z)) ).

% of_int_diff
tff(fact_5103_of__int__diff,axiom,
    ! [W: int,Z: int] : aa_int_int(ring_1_of_int_int,aa_int_int(minus_minus_int(W),Z)) = aa_int_int(minus_minus_int(aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Z)) ).

% of_int_diff
tff(fact_5104_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Xc) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W) )
    <=> ( Xc = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5105_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Xc) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W) )
    <=> ( Xc = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5106_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Xc) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W) )
    <=> ( Xc = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5107_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( ( ring_17405671764205052669omplex(Xc) = aa_nat_complex(power_power_complex(ring_17405671764205052669omplex(B2)),W) )
    <=> ( Xc = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5108_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( ( ring_18347121197199848620nteger(Xc) = power_8256067586552552935nteger(ring_18347121197199848620nteger(B2),W) )
    <=> ( Xc = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_5109_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W) = aa_int_rat(ring_1_of_int_rat,Xc) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xc ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5110_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W) = aa_int_real(ring_1_of_int_real,Xc) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xc ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5111_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W) = aa_int_int(ring_1_of_int_int,Xc) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xc ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5112_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( ( aa_nat_complex(power_power_complex(ring_17405671764205052669omplex(B2)),W) = ring_17405671764205052669omplex(Xc) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xc ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5113_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( ( power_8256067586552552935nteger(ring_18347121197199848620nteger(B2),W) = ring_18347121197199848620nteger(Xc) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xc ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_5114_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Z)),Nb) ).

% of_int_power
tff(fact_5115_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Z)),Nb) ).

% of_int_power
tff(fact_5116_of__int__power,axiom,
    ! [Z: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Z)),Nb) ).

% of_int_power
tff(fact_5117_of__int__power,axiom,
    ! [Z: int,Nb: nat] : ring_17405671764205052669omplex(aa_nat_int(power_power_int(Z),Nb)) = aa_nat_complex(power_power_complex(ring_17405671764205052669omplex(Z)),Nb) ).

% of_int_power
tff(fact_5118_of__int__power,axiom,
    ! [Z: int,Nb: nat] : ring_18347121197199848620nteger(aa_nat_int(power_power_int(Z),Nb)) = power_8256067586552552935nteger(ring_18347121197199848620nteger(Z),Nb) ).

% of_int_power
tff(fact_5119_ceiling__add__of__int,axiom,
    ! [Xc: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(plus_plus_rat(Xc),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(plus_plus_int(aa_rat_int(archim2889992004027027881ng_rat,Xc)),Z) ).

% ceiling_add_of_int
tff(fact_5120_ceiling__add__of__int,axiom,
    ! [Xc: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(plus_plus_int(aa_real_int(archim7802044766580827645g_real,Xc)),Z) ).

% ceiling_add_of_int
tff(fact_5121_ceiling__diff__of__int,axiom,
    ! [Xc: rat,Z: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(Xc),aa_int_rat(ring_1_of_int_rat,Z))) = aa_int_int(minus_minus_int(aa_rat_int(archim2889992004027027881ng_rat,Xc)),Z) ).

% ceiling_diff_of_int
tff(fact_5122_ceiling__diff__of__int,axiom,
    ! [Xc: real,Z: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),aa_int_real(ring_1_of_int_real,Z))) = aa_int_int(minus_minus_int(aa_real_int(archim7802044766580827645g_real,Xc)),Z) ).

% ceiling_diff_of_int
tff(fact_5123_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_5124_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_5125_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) ) ).

% of_int_0_le_iff
tff(fact_5126_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5127_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5128_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_5129_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5130_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5131_of__int__le__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_5132_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5133_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5134_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_le_iff
tff(fact_5135_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5136_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5137_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_5138_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_5139_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_5140_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% of_int_0_less_iff
tff(fact_5141_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Nb)),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5142_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5143_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Z) ) ).

% of_int_numeral_less_iff
tff(fact_5144_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5145_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5146_of__int__less__numeral__iff,axiom,
    ! [Z: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_5147_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5148_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5149_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_5150_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_5151_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_5152_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Z) ) ).

% of_int_1_le_iff
tff(fact_5153_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_5154_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_5155_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% of_int_1_less_iff
tff(fact_5156_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5157_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5158_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Z)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Z),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_5159_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: int,Xc: num,Nb: nat] :
      ( ( ring_18347121197199848620nteger(Ya) = power_8256067586552552935nteger(numera6620942414471956472nteger(Xc),Nb) )
    <=> ( Ya = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5160_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: int,Xc: num,Nb: nat] :
      ( ( ring_17405671764205052669omplex(Ya) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xc)),Nb) )
    <=> ( Ya = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5161_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: int,Xc: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Ya) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xc)),Nb) )
    <=> ( Ya = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5162_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: int,Xc: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Ya) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xc)),Nb) )
    <=> ( Ya = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5163_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: int,Xc: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Ya) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) )
    <=> ( Ya = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_5164_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: int] :
      ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(Xc),Nb) = ring_18347121197199848620nteger(Ya) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5165_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xc)),Nb) = ring_17405671764205052669omplex(Ya) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5166_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xc)),Nb) = aa_int_real(ring_1_of_int_real,Ya) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5167_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xc)),Nb) = aa_int_rat(ring_1_of_int_rat,Ya) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5168_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) = aa_int_int(ring_1_of_int_int,Ya) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_5169_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W)),aa_int_real(ring_1_of_int_real,Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W)),Xc) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5170_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(ring_18347121197199848620nteger(B2),W)),ring_18347121197199848620nteger(Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W)),Xc) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5171_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W)),aa_int_rat(ring_1_of_int_rat,Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W)),Xc) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5172_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W)),aa_int_int(ring_1_of_int_int,Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W)),Xc) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_5173_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Xc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5174_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ring_18347121197199848620nteger(Xc)),power_8256067586552552935nteger(ring_18347121197199848620nteger(B2),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5175_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Xc)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5176_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Xc)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_5177_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ring_18347121197199848620nteger(Xc)),power_8256067586552552935nteger(ring_18347121197199848620nteger(B2),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5178_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5179_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xc)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5180_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xc: int,B2: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xc)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_5181_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(ring_18347121197199848620nteger(B2),W)),ring_18347121197199848620nteger(Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W)),Xc) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5182_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W)),aa_int_real(ring_1_of_int_real,Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W)),Xc) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5183_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W)),aa_int_rat(ring_1_of_int_rat,Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W)),Xc) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5184_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W)),aa_int_int(ring_1_of_int_int,Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W)),Xc) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_5185_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ring_18347121197199848620nteger(A2)),power_8256067586552552935nteger(numera6620942414471956472nteger(Xc),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5186_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xc)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5187_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xc)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5188_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_5189_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(numera6620942414471956472nteger(Xc),Nb)),ring_18347121197199848620nteger(A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),A2) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5190_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xc)),Nb)),aa_int_real(ring_1_of_int_real,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),A2) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5191_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xc)),Nb)),aa_int_rat(ring_1_of_int_rat,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),A2) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5192_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),aa_int_int(ring_1_of_int_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),A2) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_5193_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ring_18347121197199848620nteger(A2)),power_8256067586552552935nteger(numera6620942414471956472nteger(Xc),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5194_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xc)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5195_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xc)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5196_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_5197_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(numera6620942414471956472nteger(Xc),Nb)),ring_18347121197199848620nteger(A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),A2) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5198_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xc)),Nb)),aa_int_real(ring_1_of_int_real,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),A2) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5199_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xc)),Nb)),aa_int_rat(ring_1_of_int_rat,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),A2) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5200_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),aa_int_int(ring_1_of_int_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),A2) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_5201_ex__le__of__int,axiom,
    ! [Xc: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_le_of_int
tff(fact_5202_ex__le__of__int,axiom,
    ! [Xc: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_le_of_int
tff(fact_5203_ex__less__of__int,axiom,
    ! [Xc: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_int_real(ring_1_of_int_real,Z2)) ).

% ex_less_of_int
tff(fact_5204_ex__less__of__int,axiom,
    ! [Xc: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),aa_int_rat(ring_1_of_int_rat,Z2)) ).

% ex_less_of_int
tff(fact_5205_ex__of__int__less,axiom,
    ! [Xc: real] :
    ? [Z2: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z2)),Xc) ).

% ex_of_int_less
tff(fact_5206_ex__of__int__less,axiom,
    ! [Xc: rat] :
    ? [Z2: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xc) ).

% ex_of_int_less
tff(fact_5207_mult__of__int__commute,axiom,
    ! [Xc: int,Ya: complex] : aa_complex_complex(times_times_complex(ring_17405671764205052669omplex(Xc)),Ya) = aa_complex_complex(times_times_complex(Ya),ring_17405671764205052669omplex(Xc)) ).

% mult_of_int_commute
tff(fact_5208_mult__of__int__commute,axiom,
    ! [Xc: int,Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Xc)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),aa_int_real(ring_1_of_int_real,Xc)) ).

% mult_of_int_commute
tff(fact_5209_mult__of__int__commute,axiom,
    ! [Xc: int,Ya: rat] : aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,Xc)),Ya) = aa_rat_rat(times_times_rat(Ya),aa_int_rat(ring_1_of_int_rat,Xc)) ).

% mult_of_int_commute
tff(fact_5210_mult__of__int__commute,axiom,
    ! [Xc: int,Ya: int] : aa_int_int(times_times_int(aa_int_int(ring_1_of_int_int,Xc)),Ya) = aa_int_int(times_times_int(Ya),aa_int_int(ring_1_of_int_int,Xc)) ).

% mult_of_int_commute
tff(fact_5211_le__of__int__ceiling,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xc))) ).

% le_of_int_ceiling
tff(fact_5212_le__of__int__ceiling,axiom,
    ! [Xc: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xc))) ).

% le_of_int_ceiling
tff(fact_5213_ceiling__le,axiom,
    ! [Xc: real,A2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_int_real(ring_1_of_int_real,A2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xc)),A2) ) ).

% ceiling_le
tff(fact_5214_ceiling__le,axiom,
    ! [Xc: rat,A2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_int_rat(ring_1_of_int_rat,A2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),A2) ) ).

% ceiling_le
tff(fact_5215_ceiling__le__iff,axiom,
    ! [Xc: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xc)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_int_real(ring_1_of_int_real,Z)) ) ).

% ceiling_le_iff
tff(fact_5216_ceiling__le__iff,axiom,
    ! [Xc: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% ceiling_le_iff
tff(fact_5217_less__ceiling__iff,axiom,
    ! [Z: int,Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_rat_int(archim2889992004027027881ng_rat,Xc))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xc) ) ).

% less_ceiling_iff
tff(fact_5218_less__ceiling__iff,axiom,
    ! [Z: int,Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_real_int(archim7802044766580827645g_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),Xc) ) ).

% less_ceiling_iff
tff(fact_5219_real__of__int__div4,axiom,
    ! [Nb: int,Xc: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xc))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xc))) ).

% real_of_int_div4
tff(fact_5220_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_nonneg
tff(fact_5221_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_nonneg
tff(fact_5222_of__int__nonneg,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_nonneg
tff(fact_5223_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Z)) ) ).

% of_int_pos
tff(fact_5224_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Z)) ) ).

% of_int_pos
tff(fact_5225_of__int__pos,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Z)) ) ).

% of_int_pos
tff(fact_5226_floor__exists,axiom,
    ! [Xc: real] :
    ? [Z2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z2)),Xc)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(Z2),one_one_int))) ) ).

% floor_exists
tff(fact_5227_floor__exists,axiom,
    ! [Xc: rat] :
    ? [Z2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z2)),Xc)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(Z2),one_one_int))) ) ).

% floor_exists
tff(fact_5228_floor__exists1,axiom,
    ! [Xc: real] :
    ? [X3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X3)),Xc)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(X3),one_one_int)))
      & ! [Y3: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y3)),Xc)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(Y3),one_one_int))) )
         => ( Y3 = X3 ) ) ) ).

% floor_exists1
tff(fact_5229_floor__exists1,axiom,
    ! [Xc: rat] :
    ? [X3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X3)),Xc)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(X3),one_one_int)))
      & ! [Y3: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y3)),Xc)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(Y3),one_one_int))) )
         => ( Y3 = X3 ) ) ) ).

% floor_exists1
tff(fact_5230_of__int__ceiling__le__add__one,axiom,
    ! [R3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R3))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R3),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_5231_of__int__ceiling__le__add__one,axiom,
    ! [R3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R3))),aa_rat_rat(plus_plus_rat(R3),one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_5232_of__int__ceiling__diff__one__le,axiom,
    ! [R3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R3))),one_one_real)),R3) ).

% of_int_ceiling_diff_one_le
tff(fact_5233_of__int__ceiling__diff__one__le,axiom,
    ! [R3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R3))),one_one_rat)),R3) ).

% of_int_ceiling_diff_one_le
tff(fact_5234_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xc: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xc) ) ).

% of_nat_less_of_int_iff
tff(fact_5235_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xc: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_int_real(ring_1_of_int_real,Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xc) ) ).

% of_nat_less_of_int_iff
tff(fact_5236_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(ring_1_of_int_int,Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xc) ) ).

% of_nat_less_of_int_iff
tff(fact_5237_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xc: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)),ring_18347121197199848620nteger(Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xc) ) ).

% of_nat_less_of_int_iff
tff(fact_5238_int__le__real__less,axiom,
    ! [Nb: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,M)),one_one_real)) ) ).

% int_le_real_less
tff(fact_5239_int__less__real__le,axiom,
    ! [Nb: int,M: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),M)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real)),aa_int_real(ring_1_of_int_real,M)) ) ).

% int_less_real_le
tff(fact_5240_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2)),Xc) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_5241_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] : aa_nat_nat(vEBT_T_p_r_e_d(aa_o_VEBT_VEBT(vEBT_Leaf((Uu)),(Uv2))),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_5242_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw: $o,Nb: nat] : aa_nat_nat(vEBT_T_s_u_c_c(aa_o_VEBT_VEBT(vEBT_Leaf((Uv2)),(Uw))),aa_nat_nat(suc,Nb)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_5243_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz2: list_VEBT_VEBT,Vaa: vEBT_VEBT,Vb2: nat] : aa_nat_nat(vEBT_T_p_r_e_d(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Vaa)),Vb2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_5244_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,Ta))
    <=> ! [I2: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I2)),one_one_real)),Ta)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),aa_int_real(ring_1_of_int_real,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_5245_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: rat] :
      ( aa_int_o(P,aa_rat_int(archim2889992004027027881ng_rat,Ta))
    <=> ! [I2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I2)),one_one_rat)),Ta)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),aa_int_rat(ring_1_of_int_rat,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_5246_ceiling__eq__iff,axiom,
    ! [Xc: real,A2: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,Xc) = A2 )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A2)),one_one_real)),Xc)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_int_real(ring_1_of_int_real,A2)) ) ) ).

% ceiling_eq_iff
tff(fact_5247_ceiling__eq__iff,axiom,
    ! [Xc: rat,A2: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,Xc) = A2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A2)),one_one_rat)),Xc)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_int_rat(ring_1_of_int_rat,A2)) ) ) ).

% ceiling_eq_iff
tff(fact_5248_ceiling__unique,axiom,
    ! [Z: int,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_int_real(ring_1_of_int_real,Z))
       => ( aa_real_int(archim7802044766580827645g_real,Xc) = Z ) ) ) ).

% ceiling_unique
tff(fact_5249_ceiling__unique,axiom,
    ! [Z: int,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_int_rat(ring_1_of_int_rat,Z))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xc) = Z ) ) ) ).

% ceiling_unique
tff(fact_5250_ceiling__correct,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xc))),one_one_real)),Xc)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xc))) ) ).

% ceiling_correct
tff(fact_5251_ceiling__correct,axiom,
    ! [Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xc))),one_one_rat)),Xc)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xc))) ) ).

% ceiling_correct
tff(fact_5252_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Vaa: nat] : aa_nat_nat(vEBT_T_s_u_c_c(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Vaa) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_5253_ceiling__less__iff,axiom,
    ! [Xc: real,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xc)),Z)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_5254_ceiling__less__iff,axiom,
    ! [Xc: rat,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),Z)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_5255_le__ceiling__iff,axiom,
    ! [Z: int,Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_rat_int(archim2889992004027027881ng_rat,Xc))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Z)),one_one_rat)),Xc) ) ).

% le_ceiling_iff
tff(fact_5256_le__ceiling__iff,axiom,
    ! [Z: int,Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),aa_real_int(archim7802044766580827645g_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Z)),one_one_real)),Xc) ) ).

% le_ceiling_iff
tff(fact_5257_real__of__int__div2,axiom,
    ! [Nb: int,Xc: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xc))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xc)))) ).

% real_of_int_div2
tff(fact_5258_real__of__int__div3,axiom,
    ! [Nb: int,Xc: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xc))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xc)))),one_one_real) ).

% real_of_int_div3
tff(fact_5259_ceiling__divide__upper,axiom,
    ! [Q3: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,P3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P3),Q3)))),Q3)) ) ).

% ceiling_divide_upper
tff(fact_5260_ceiling__divide__upper,axiom,
    ! [Q3: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P3),aa_rat_rat(times_times_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P3),Q3)))),Q3)) ) ).

% ceiling_divide_upper
tff(fact_5261_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)),Xc) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_5262_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Vaa: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_5263_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xc: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),Xc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(Xc = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_5264_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : aa_nat_nat(vEBT_T_p_r_e_d(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd2),Ve2)),Vf2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_5265_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] : aa_nat_nat(vEBT_T_s_u_c_c(aa_o_VEBT_VEBT(vEBT_Leaf((Uu)),(B2))),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_5266_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : aa_nat_nat(vEBT_T_s_u_c_c(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd2)),Ve2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_5267_ceiling__divide__lower,axiom,
    ! [Q3: real,P3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q3)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P3),Q3)))),one_one_real)),Q3)),P3) ) ).

% ceiling_divide_lower
tff(fact_5268_ceiling__divide__lower,axiom,
    ! [Q3: rat,P3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q3)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P3),Q3)))),one_one_rat)),Q3)),P3) ) ).

% ceiling_divide_lower
tff(fact_5269_ceiling__eq,axiom,
    ! [Nb: int,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,Xc) = aa_int_int(plus_plus_int(Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_5270_ceiling__eq,axiom,
    ! [Nb: int,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Nb)),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_rat_rat(plus_plus_rat(aa_int_rat(ring_1_of_int_rat,Nb)),one_one_rat))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xc) = aa_int_int(plus_plus_int(Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_5271_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] : aa_nat_nat(vEBT_T_p_r_e_d(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(Uw))),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_5272_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(vEBT_T_p_r_e_d(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_5273_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(vEBT_T_s_u_c_c(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_5274_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista),Summarya)),Xc) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_5275_insersimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Ya: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),X_12)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(Ta),Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insersimp
tff(fact_5276_insertsimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(Ta),L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ) ) ).

% insertsimp
tff(fact_5277_insert__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(Ta),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))))) ) ).

% insert_bound_height
tff(fact_5278_norm__divide__numeral,axiom,
    ! [A2: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_5279_norm__divide__numeral,axiom,
    ! [A2: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A2),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_5280_norm__mult__numeral2,axiom,
    ! [A2: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_5281_norm__mult__numeral2,axiom,
    ! [A2: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(A2),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_5282_norm__mult__numeral1,axiom,
    ! [W: num,A2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(real_V7735802525324610683m_real,A2)) ).

% norm_mult_numeral1
tff(fact_5283_norm__mult__numeral1,axiom,
    ! [W: num,A2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_complex_real(real_V1022390504157884413omplex,A2)) ).

% norm_mult_numeral1
tff(fact_5284_norm__le__zero__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xc)),zero_zero_real)
    <=> ( Xc = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_5285_norm__le__zero__iff,axiom,
    ! [Xc: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),zero_zero_real)
    <=> ( Xc = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_5286_zero__less__norm__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,Xc))
    <=> ( Xc != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_5287_zero__less__norm__iff,axiom,
    ! [Xc: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xc))
    <=> ( Xc != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_5288_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_5289_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_5290_norm__eq__zero,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(real_V7735802525324610683m_real,Xc) = zero_zero_real )
    <=> ( Xc = zero_zero_real ) ) ).

% norm_eq_zero
tff(fact_5291_norm__eq__zero,axiom,
    ! [Xc: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Xc) = zero_zero_real )
    <=> ( Xc = zero_zero_complex ) ) ).

% norm_eq_zero
tff(fact_5292_norm__zero,axiom,
    aa_real_real(real_V7735802525324610683m_real,zero_zero_real) = zero_zero_real ).

% norm_zero
tff(fact_5293_norm__zero,axiom,
    aa_complex_real(real_V1022390504157884413omplex,zero_zero_complex) = zero_zero_real ).

% norm_zero
tff(fact_5294_norm__minus__commute,axiom,
    ! [A2: real,B2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)) ).

% norm_minus_commute
tff(fact_5295_norm__minus__commute,axiom,
    ! [A2: complex,B2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A2),B2)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B2),A2)) ).

% norm_minus_commute
tff(fact_5296_norm__mult,axiom,
    ! [Xc: real,Ya: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xc)),aa_real_real(real_V7735802525324610683m_real,Ya)) ).

% norm_mult
tff(fact_5297_norm__mult,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),aa_complex_real(real_V1022390504157884413omplex,Ya)) ).

% norm_mult
tff(fact_5298_norm__not__less__zero,axiom,
    ! [Xc: complex] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),zero_zero_real) ).

% norm_not_less_zero
tff(fact_5299_norm__ge__zero,axiom,
    ! [Xc: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xc)) ).

% norm_ge_zero
tff(fact_5300_norm__divide,axiom,
    ! [A2: real,B2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2)) ).

% norm_divide
tff(fact_5301_norm__divide,axiom,
    ! [A2: complex,B2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2)) ).

% norm_divide
tff(fact_5302_norm__power,axiom,
    ! [Xc: real,Nb: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xc)),Nb) ).

% norm_power
tff(fact_5303_norm__power,axiom,
    ! [Xc: complex,Nb: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xc),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),Nb) ).

% norm_power
tff(fact_5304_power__eq__imp__eq__norm,axiom,
    ! [W: real,Nb: nat,Z: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5305_power__eq__imp__eq__norm,axiom,
    ! [W: complex,Nb: nat,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),Nb) = aa_nat_complex(power_power_complex(Z),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Z) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_5306_nonzero__norm__divide,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_5307_nonzero__norm__divide,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_5308_norm__mult__less,axiom,
    ! [Xc: real,R3: real,Ya: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xc)),R3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Ya)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),S2)) ) ) ).

% norm_mult_less
tff(fact_5309_norm__mult__less,axiom,
    ! [Xc: complex,R3: real,Ya: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),R3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Ya)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xc),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),S2)) ) ) ).

% norm_mult_less
tff(fact_5310_norm__mult__ineq,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xc)),aa_real_real(real_V7735802525324610683m_real,Ya))) ).

% norm_mult_ineq
tff(fact_5311_norm__mult__ineq,axiom,
    ! [Xc: complex,Ya: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xc),Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),aa_complex_real(real_V1022390504157884413omplex,Ya))) ).

% norm_mult_ineq
tff(fact_5312_norm__triangle__lt,axiom,
    ! [Xc: real,Ya: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xc)),aa_real_real(real_V7735802525324610683m_real,Ya))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya))),E) ) ).

% norm_triangle_lt
tff(fact_5313_norm__triangle__lt,axiom,
    ! [Xc: complex,Ya: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),aa_complex_real(real_V1022390504157884413omplex,Ya))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xc),Ya))),E) ) ).

% norm_triangle_lt
tff(fact_5314_norm__add__less,axiom,
    ! [Xc: real,R3: real,Ya: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xc)),R3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Ya)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R3),S2)) ) ) ).

% norm_add_less
tff(fact_5315_norm__add__less,axiom,
    ! [Xc: complex,R3: real,Ya: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),R3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Ya)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xc),Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R3),S2)) ) ) ).

% norm_add_less
tff(fact_5316_norm__add__leD,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))),C2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A2)),C2)) ) ).

% norm_add_leD
tff(fact_5317_norm__add__leD,axiom,
    ! [A2: complex,B2: complex,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A2),B2))),C2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),C2)) ) ).

% norm_add_leD
tff(fact_5318_norm__triangle__le,axiom,
    ! [Xc: real,Ya: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xc)),aa_real_real(real_V7735802525324610683m_real,Ya))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya))),E) ) ).

% norm_triangle_le
tff(fact_5319_norm__triangle__le,axiom,
    ! [Xc: complex,Ya: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),aa_complex_real(real_V1022390504157884413omplex,Ya))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xc),Ya))),E) ) ).

% norm_triangle_le
tff(fact_5320_norm__triangle__ineq,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xc)),aa_real_real(real_V7735802525324610683m_real,Ya))) ).

% norm_triangle_ineq
tff(fact_5321_norm__triangle__ineq,axiom,
    ! [Xc: complex,Ya: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xc),Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),aa_complex_real(real_V1022390504157884413omplex,Ya))) ).

% norm_triangle_ineq
tff(fact_5322_norm__triangle__mono,axiom,
    ! [A2: real,R3: real,B2: real,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A2)),R3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B2)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R3),S2)) ) ) ).

% norm_triangle_mono
tff(fact_5323_norm__triangle__mono,axiom,
    ! [A2: complex,R3: real,B2: complex,S2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A2)),R3)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B2)),S2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R3),S2)) ) ) ).

% norm_triangle_mono
tff(fact_5324_norm__diff__triangle__less,axiom,
    ! [Xc: real,Ya: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Ya))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_5325_norm__diff__triangle__less,axiom,
    ! [Xc: complex,Ya: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xc),Ya))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Ya),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xc),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_5326_norm__power__ineq,axiom,
    ! [Xc: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xc)),Nb)) ).

% norm_power_ineq
tff(fact_5327_norm__power__ineq,axiom,
    ! [Xc: complex,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xc),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),Nb)) ).

% norm_power_ineq
tff(fact_5328_norm__triangle__sub,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Ya)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Ya)))) ).

% norm_triangle_sub
tff(fact_5329_norm__triangle__sub,axiom,
    ! [Xc: complex,Ya: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Ya)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xc),Ya)))) ).

% norm_triangle_sub
tff(fact_5330_norm__triangle__ineq4,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2))) ).

% norm_triangle_ineq4
tff(fact_5331_norm__triangle__ineq4,axiom,
    ! [A2: complex,B2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2))) ).

% norm_triangle_ineq4
tff(fact_5332_norm__diff__triangle__le,axiom,
    ! [Xc: real,Ya: real,E1: real,Z: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Ya))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_5333_norm__diff__triangle__le,axiom,
    ! [Xc: complex,Ya: complex,E1: real,Z: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xc),Ya))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Ya),Z))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xc),Z))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_5334_norm__triangle__le__diff,axiom,
    ! [Xc: real,Ya: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xc)),aa_real_real(real_V7735802525324610683m_real,Ya))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Ya))),E) ) ).

% norm_triangle_le_diff
tff(fact_5335_norm__triangle__le__diff,axiom,
    ! [Xc: complex,Ya: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),aa_complex_real(real_V1022390504157884413omplex,Ya))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xc),Ya))),E) ) ).

% norm_triangle_le_diff
tff(fact_5336_norm__diff__ineq,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))) ).

% norm_diff_ineq
tff(fact_5337_norm__diff__ineq,axiom,
    ! [A2: complex,B2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A2),B2))) ).

% norm_diff_ineq
tff(fact_5338_norm__triangle__ineq2,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))) ).

% norm_triangle_ineq2
tff(fact_5339_norm__triangle__ineq2,axiom,
    ! [A2: complex,B2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A2),B2))) ).

% norm_triangle_ineq2
tff(fact_5340_power__eq__1__iff,axiom,
    ! [W: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5341_power__eq__1__iff,axiom,
    ! [W: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(W),Nb) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_5342_norm__diff__triangle__ineq,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),D2)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D2)))) ).

% norm_diff_triangle_ineq
tff(fact_5343_norm__diff__triangle__ineq,axiom,
    ! [A2: complex,B2: complex,C2: complex,D2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A2),B2)),aa_complex_complex(plus_plus_complex(C2),D2)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A2),C2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B2),D2)))) ).

% norm_diff_triangle_ineq
tff(fact_5344_square__norm__one,axiom,
    ! [Xc: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,Xc) = one_one_real ) ) ).

% square_norm_one
tff(fact_5345_square__norm__one,axiom,
    ! [Xc: complex] :
      ( ( aa_nat_complex(power_power_complex(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,Xc) = one_one_real ) ) ).

% square_norm_one
tff(fact_5346_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),W)))) ) ) ).

% norm_power_diff
tff(fact_5347_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Z),M)),aa_nat_complex(power_power_complex(W),M)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Z),W)))) ) ) ).

% norm_power_diff
tff(fact_5348_del__in__range,axiom,
    ! [Mia: nat,Xc: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xc)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = $let(
                xn: nat,
                xn:= 
                  $ite(Xc = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(vEBT_vebt_mint(Summarya)))))),Xc),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xc = Mia,xn,Mia),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                      $let(
                        newnode2: vEBT_VEBT,
                        newnode2:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode2),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode2),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summarya),h),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                      $ite(
                                        xn = Maa,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                        Maa ))),Deg,newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                    $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h)))),Maa))),Deg,newlist),Summarya) ) ) ),
                      aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_5349_del__x__mi,axiom,
    ! [Xc: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat] :
      ( ( ( Xc = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = $let(
                        newnode2: vEBT_VEBT,
                        newnode2:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L),
                        $let(
                          newlist: list_VEBT_VEBT,
                          newlist:= list_u1324408373059187874T_VEBT(TreeLista,H,newnode2),
                          $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode2),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summarya),H),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                      $ite(
                                        Xn = Maa,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                        Maa ))),Deg,newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),H)))),Maa))),Deg,newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_5350_del__x__mi__lets__in,axiom,
    ! [Xc: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnodea: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( Xc = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnodea = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnodea) )
                     => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = $ite(
                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnodea),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summarya),H),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                      $ite(
                                        Xn = Maa,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(maxs)))))) ),
                                        Maa ))),Deg,Newlist),sn) ),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                    $ite(Xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H)))),Maa))),Deg,Newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_5351_del__x__mi__lets__in__minNull,axiom,
    ! [Xc: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list_VEBT_VEBT,L: nat,Newnodea: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( Xc = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( Xn = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(vEBT_vebt_mint(Summarya)))))) )
             => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                 => ( ( Newnodea = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnodea) )
                     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnodea)
                       => ( ( Sn = aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summarya),H) )
                         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xn),
                                      $ite(
                                        Xn = Maa,
                                        $let(
                                          maxs: option_nat,
                                          maxs:= vEBT_vebt_maxt(Sn),
                                          $ite(maxs = none_nat,Xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(maxs)))))) ),
                                        Maa ))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_5352_del__x__mia,axiom,
    ! [Xc: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xc = Mia )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = $let(
                xn: nat,
                xn:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(vEBT_vebt_mint(Summarya)))))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(
                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                    $let(
                      newnode2: vEBT_VEBT,
                      newnode2:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode2),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode2),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summarya),h),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                    $ite(
                                      xn = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,xn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                      Maa ))),Deg,newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(xn),
                                  $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h)))),Maa))),Deg,newlist),Summarya) ) ) ),
                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_5353_del__x__not__mi,axiom,
    ! [Mia: nat,Xc: nat,Maa: nat,Deg: nat,H: nat,L: nat,Newnodea: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xc)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnodea = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L) )
               => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnodea) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                   => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnodea),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summarya),H),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                    $ite(
                                      Xc = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(maxs)))))) ),
                                      Maa ))),Deg,Newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                  $ite(Xc = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),H)))),Maa))),Deg,Newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_5354_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Ta)) ).

% set_vebt'_def
tff(fact_5355_succ__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_succ(Ta,Xc) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_af(Ta),Xc)) = bot_bot_set_nat ) ) ) ).

% succ_empty
tff(fact_5356_pred__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( vEBT_vebt_pred(Ta,Xc) = none_nat )
      <=> ( collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ag(Ta),Xc)) = bot_bot_set_nat ) ) ) ).

% pred_empty
tff(fact_5357_singleton__conv2,axiom,
    ! [A2: vEBT_VEBT] : collect_VEBT_VEBT(fequal_VEBT_VEBT(A2)) = insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv2
tff(fact_5358_singleton__conv2,axiom,
    ! [A2: complex] : collect_complex(fequal_complex(A2)) = insert_complex(A2,bot_bot_set_complex) ).

% singleton_conv2
tff(fact_5359_singleton__conv2,axiom,
    ! [A2: product_prod_int_int] : collec213857154873943460nt_int(fequal6010348034681699726nt_int(A2)) = insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int) ).

% singleton_conv2
tff(fact_5360_singleton__conv2,axiom,
    ! [A2: real] : collect_real(aa_real_fun_real_o(fequal_real,A2)) = insert_real(A2,bot_bot_set_real) ).

% singleton_conv2
tff(fact_5361_singleton__conv2,axiom,
    ! [A2: $o] : collect_o(fequal_o((A2))) = insert_o((A2),bot_bot_set_o) ).

% singleton_conv2
tff(fact_5362_singleton__conv2,axiom,
    ! [A2: nat] : collect_nat(aa_nat_fun_nat_o(fequal_nat,A2)) = insert_nat(A2,bot_bot_set_nat) ).

% singleton_conv2
tff(fact_5363_singleton__conv2,axiom,
    ! [A2: int] : collect_int(aa_int_fun_int_o(fequal_int,A2)) = insert_int(A2,bot_bot_set_int) ).

% singleton_conv2
tff(fact_5364_singleton__conv,axiom,
    ! [A2: vEBT_VEBT] : collect_VEBT_VEBT(aTP_Lamm_ah(A2)) = insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT) ).

% singleton_conv
tff(fact_5365_singleton__conv,axiom,
    ! [A2: complex] : collect_complex(aTP_Lamm_ai(A2)) = insert_complex(A2,bot_bot_set_complex) ).

% singleton_conv
tff(fact_5366_singleton__conv,axiom,
    ! [A2: product_prod_int_int] : collec213857154873943460nt_int(aTP_Lamm_aj(A2)) = insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int) ).

% singleton_conv
tff(fact_5367_singleton__conv,axiom,
    ! [A2: real] : collect_real(aa_real_fun_real_o(aTP_Lamm_ak,A2)) = insert_real(A2,bot_bot_set_real) ).

% singleton_conv
tff(fact_5368_singleton__conv,axiom,
    ! [A2: $o] : collect_o(aTP_Lamm_al((A2))) = insert_o((A2),bot_bot_set_o) ).

% singleton_conv
tff(fact_5369_singleton__conv,axiom,
    ! [A2: nat] : collect_nat(aa_nat_fun_nat_o(aTP_Lamm_am,A2)) = insert_nat(A2,bot_bot_set_nat) ).

% singleton_conv
tff(fact_5370_singleton__conv,axiom,
    ! [A2: int] : collect_int(aa_int_fun_int_o(aTP_Lamm_an,A2)) = insert_int(A2,bot_bot_set_int) ).

% singleton_conv
tff(fact_5371_del__x__not__mia,axiom,
    ! [Mia: nat,Xc: nat,Maa: nat,Deg: nat,H: nat,L: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xc)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
               => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = $let(
                      newnode2: vEBT_VEBT,
                      newnode2:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L),
                      $let(
                        newlist: list_VEBT_VEBT,
                        newlist:= list_u1324408373059187874T_VEBT(TreeLista,H,newnode2),
                        $ite(
                          aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode2),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summarya),H),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                    $ite(
                                      Xc = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                      Maa ))),Deg,newlist),sn) ),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                  $ite(Xc = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,H),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),H)))),Maa))),Deg,newlist),Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_5372_del__x__not__mi__new__node__nil,axiom,
    ! [Mia: nat,Xc: nat,Maa: nat,Deg: nat,H: nat,L: nat,Newnodea: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Sn: vEBT_VEBT,Summarya: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xc)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Maa) )
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = H )
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = L )
             => ( ( Newnodea = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),H)),L) )
               => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Newnodea)
                 => ( ( Sn = aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summarya),H) )
                   => ( ( Newlist = list_u1324408373059187874T_VEBT(TreeLista,H,Newnodea) )
                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
                       => ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya)),Xc) = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),
                                    $ite(
                                      Xc = Maa,
                                      $let(
                                        maxs: option_nat,
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none_nat,Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Newlist),the_nat(maxs)))))) ),
                                      Maa ))),Deg,Newlist),Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_5373_vebt__memberi_H__rf__abstr,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xc: nat] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_V854960066525838166emberi(Ta,Tib,Xc),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_ao(Ta),Tib),Xc)) ).

% vebt_memberi'_rf_abstr
tff(fact_5374_vebt__inserti_H__rf__abstr,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xc: nat] : hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_V3964819847710782039nserti(Ta,Tib,Xc),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Ta),Xc))) ).

% vebt_inserti'_rf_abstr
tff(fact_5375_vebt__succi_H__rf__abstr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_VEBT_vebt_succi(Ta,Tib,Xc),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_ap(Ta),Tib),Xc)) ) ).

% vebt_succi'_rf_abstr
tff(fact_5376_vebt__pred_H__rf__abstr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_VEBT_vebt_predi(Ta,Tib,Xc),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_aq(Ta),Tib),Xc)) ) ).

% vebt_pred'_rf_abstr
tff(fact_5377_Collect__conv__if,axiom,
    ! [A2: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_ar(A2),P)) = $ite(aa_VEBT_VEBT_o(P,A2),insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ).

% Collect_conv_if
tff(fact_5378_Collect__conv__if,axiom,
    ! [A2: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_as(A2),P)) = $ite(aa_complex_o(P,A2),insert_complex(A2,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if
tff(fact_5379_Collect__conv__if,axiom,
    ! [A2: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_at(A2),P)) = $ite(aa_Pro4592068669816069595_int_o(P,A2),insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int),bot_bo1796632182523588997nt_int) ).

% Collect_conv_if
tff(fact_5380_Collect__conv__if,axiom,
    ! [A2: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_au(A2),P)) = $ite(aa_real_o(P,A2),insert_real(A2,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if
tff(fact_5381_Collect__conv__if,axiom,
    ! [A2: $o,P: fun_o_o] :
      collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_av((A2)),P)) = $ite(aa_o_o(P,(A2)),insert_o((A2),bot_bot_set_o),bot_bot_set_o) ).

% Collect_conv_if
tff(fact_5382_Collect__conv__if,axiom,
    ! [A2: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_aw(A2),P)) = $ite(aa_nat_o(P,A2),insert_nat(A2,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if
tff(fact_5383_Collect__conv__if,axiom,
    ! [A2: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_ax(A2),P)) = $ite(aa_int_o(P,A2),insert_int(A2,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if
tff(fact_5384_Collect__conv__if2,axiom,
    ! [A2: vEBT_VEBT,P: fun_VEBT_VEBT_o] :
      collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_ay(A2),P)) = $ite(aa_VEBT_VEBT_o(P,A2),insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT),bot_bo8194388402131092736T_VEBT) ).

% Collect_conv_if2
tff(fact_5385_Collect__conv__if2,axiom,
    ! [A2: complex,P: fun_complex_o] :
      collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_az(A2),P)) = $ite(aa_complex_o(P,A2),insert_complex(A2,bot_bot_set_complex),bot_bot_set_complex) ).

% Collect_conv_if2
tff(fact_5386_Collect__conv__if2,axiom,
    ! [A2: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] :
      collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_ba(A2),P)) = $ite(aa_Pro4592068669816069595_int_o(P,A2),insert5033312907999012233nt_int(A2,bot_bo1796632182523588997nt_int),bot_bo1796632182523588997nt_int) ).

% Collect_conv_if2
tff(fact_5387_Collect__conv__if2,axiom,
    ! [A2: real,P: fun_real_o] :
      collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bb(A2),P)) = $ite(aa_real_o(P,A2),insert_real(A2,bot_bot_set_real),bot_bot_set_real) ).

% Collect_conv_if2
tff(fact_5388_Collect__conv__if2,axiom,
    ! [A2: $o,P: fun_o_o] :
      collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_bc((A2)),P)) = $ite(aa_o_o(P,(A2)),insert_o((A2),bot_bot_set_o),bot_bot_set_o) ).

% Collect_conv_if2
tff(fact_5389_Collect__conv__if2,axiom,
    ! [A2: nat,P: fun_nat_o] :
      collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bd(A2),P)) = $ite(aa_nat_o(P,A2),insert_nat(A2,bot_bot_set_nat),bot_bot_set_nat) ).

% Collect_conv_if2
tff(fact_5390_Collect__conv__if2,axiom,
    ! [A2: int,P: fun_int_o] :
      collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_be(A2),P)) = $ite(aa_int_o(P,A2),insert_int(A2,bot_bot_set_int),bot_bot_set_int) ).

% Collect_conv_if2
tff(fact_5391_max__def__raw,axiom,
    ! [X4: set_nat,Xa: set_nat] :
      ord_max_set_nat(X4,Xa) = $ite(aa_set_nat_o(ord_less_eq_set_nat(X4),Xa),Xa,X4) ).

% max_def_raw
tff(fact_5392_max__def__raw,axiom,
    ! [X4: rat,Xa: rat] :
      aa_rat_rat(ord_max_rat(X4),Xa) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Xa),Xa,X4) ).

% max_def_raw
tff(fact_5393_max__def__raw,axiom,
    ! [X4: num,Xa: num] :
      aa_num_num(ord_max_num(X4),Xa) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Xa),Xa,X4) ).

% max_def_raw
tff(fact_5394_max__def__raw,axiom,
    ! [X4: nat,Xa: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X4),Xa) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa),Xa,X4) ).

% max_def_raw
tff(fact_5395_max__def__raw,axiom,
    ! [X4: int,Xa: int] :
      aa_int_int(ord_max_int(X4),Xa) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Xa),Xa,X4) ).

% max_def_raw
tff(fact_5396_Collect__subset,axiom,
    ! [A: set_real,P: fun_real_o] : aa_set_real_o(ord_less_eq_set_real(collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bf(A),P))),A) ).

% Collect_subset
tff(fact_5397_Collect__subset,axiom,
    ! [A: set_VEBT_VEBT,P: fun_VEBT_VEBT_o] : aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_bg(A),P))),A) ).

% Collect_subset
tff(fact_5398_Collect__subset,axiom,
    ! [A: set_int,P: fun_int_o] : aa_set_int_o(ord_less_eq_set_int(collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(A),P))),A) ).

% Collect_subset
tff(fact_5399_Collect__subset,axiom,
    ! [A: set_complex,P: fun_complex_o] : aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_bi(A),P))),A) ).

% Collect_subset
tff(fact_5400_Collect__subset,axiom,
    ! [A: set_Pr958786334691620121nt_int,P: fun_Pr3238374608701176162_int_o] : ord_le2843351958646193337nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_bj(A),P)),A) ).

% Collect_subset
tff(fact_5401_Collect__subset,axiom,
    ! [A: set_nat,P: fun_nat_o] : aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_bk(A),P))),A) ).

% Collect_subset
tff(fact_5402_less__eq__set__def,axiom,
    ! [A: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),B)
    <=> ord_le3605727316730466277real_o(aTP_Lamm_a(A),aTP_Lamm_a(B)) ) ).

% less_eq_set_def
tff(fact_5403_less__eq__set__def,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),B)
    <=> ord_le88303844413636359VEBT_o(aTP_Lamm_aa(A),aTP_Lamm_aa(B)) ) ).

% less_eq_set_def
tff(fact_5404_less__eq__set__def,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
    <=> ord_le5418975059635765733_int_o(aTP_Lamm_ac(A),aTP_Lamm_ac(B)) ) ).

% less_eq_set_def
tff(fact_5405_less__eq__set__def,axiom,
    ! [A: set_complex,B: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B)
    <=> ord_le7644322562001619171plex_o(aTP_Lamm_ad(A),aTP_Lamm_ad(B)) ) ).

% less_eq_set_def
tff(fact_5406_less__eq__set__def,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
    <=> ord_le7240017405068453697_nat_o(aTP_Lamm_ab(A),aTP_Lamm_ab(B)) ) ).

% less_eq_set_def
tff(fact_5407_less__set__def,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_set_nat(A),B)
    <=> ord_less_fun_nat_o(aTP_Lamm_ab(A),aTP_Lamm_ab(B)) ) ).

% less_set_def
tff(fact_5408_less__set__def,axiom,
    ! [A: set_real,B: set_real] :
      ( aa_set_real_o(ord_less_set_real(A),B)
    <=> ord_less_fun_real_o(aTP_Lamm_a(A),aTP_Lamm_a(B)) ) ).

% less_set_def
tff(fact_5409_less__set__def,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(A),B)
    <=> ord_le3534172602983796987VEBT_o(aTP_Lamm_aa(A),aTP_Lamm_aa(B)) ) ).

% less_set_def
tff(fact_5410_less__set__def,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_set_int(A),B)
    <=> ord_less_fun_int_o(aTP_Lamm_ac(A),aTP_Lamm_ac(B)) ) ).

% less_set_def
tff(fact_5411_less__set__def,axiom,
    ! [A: set_complex,B: set_complex] :
      ( aa_set_complex_o(ord_less_set_complex(A),B)
    <=> ord_le4409593196786536943plex_o(aTP_Lamm_ad(A),aTP_Lamm_ad(B)) ) ).

% less_set_def
tff(fact_5412_minus__set__def,axiom,
    ! [A: set_real,B: set_real] : minus_minus_set_real(A,B) = collect_real(minus_404528632214381484real_o(aTP_Lamm_a(A),aTP_Lamm_a(B))) ).

% minus_set_def
tff(fact_5413_minus__set__def,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A,B) = collect_VEBT_VEBT(minus_4582106842932979072VEBT_o(aTP_Lamm_aa(A),aTP_Lamm_aa(B))) ).

% minus_set_def
tff(fact_5414_minus__set__def,axiom,
    ! [A: set_int,B: set_int] : minus_minus_set_int(A,B) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ac(A),aTP_Lamm_ac(B))) ).

% minus_set_def
tff(fact_5415_minus__set__def,axiom,
    ! [A: set_complex,B: set_complex] : minus_811609699411566653omplex(A,B) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_ad(A),aTP_Lamm_ad(B))) ).

% minus_set_def
tff(fact_5416_minus__set__def,axiom,
    ! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A,B) = collec213857154873943460nt_int(minus_827889751674334651_int_o(aTP_Lamm_ae(A),aTP_Lamm_ae(B))) ).

% minus_set_def
tff(fact_5417_minus__set__def,axiom,
    ! [A: set_nat,B: set_nat] : minus_minus_set_nat(A,B) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ab(A),aTP_Lamm_ab(B))) ).

% minus_set_def
tff(fact_5418_set__diff__eq,axiom,
    ! [A: set_real,B: set_real] : minus_minus_set_real(A,B) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_bl(A),B)) ).

% set_diff_eq
tff(fact_5419_set__diff__eq,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] : minus_5127226145743854075T_VEBT(A,B) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_bm(A),B)) ).

% set_diff_eq
tff(fact_5420_set__diff__eq,axiom,
    ! [A: set_int,B: set_int] : minus_minus_set_int(A,B) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_bn(A),B)) ).

% set_diff_eq
tff(fact_5421_set__diff__eq,axiom,
    ! [A: set_complex,B: set_complex] : minus_811609699411566653omplex(A,B) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_bo(A),B)) ).

% set_diff_eq
tff(fact_5422_set__diff__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int,B: set_Pr958786334691620121nt_int] : minus_1052850069191792384nt_int(A,B) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_bp(A),B)) ).

% set_diff_eq
tff(fact_5423_set__diff__eq,axiom,
    ! [A: set_nat,B: set_nat] : minus_minus_set_nat(A,B) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_bq(A),B)) ).

% set_diff_eq
tff(fact_5424_insert__compr,axiom,
    ! [A2: $o,B: set_o] : insert_o((A2),B) = collect_o(aa_set_o_fun_o_o(aTP_Lamm_br((A2)),B)) ).

% insert_compr
tff(fact_5425_insert__compr,axiom,
    ! [A2: real,B: set_real] : insert_real(A2,B) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_bs(A2),B)) ).

% insert_compr
tff(fact_5426_insert__compr,axiom,
    ! [A2: vEBT_VEBT,B: set_VEBT_VEBT] : insert_VEBT_VEBT(A2,B) = collect_VEBT_VEBT(aa_set2497645856053385599VEBT_o(aTP_Lamm_bt(A2),B)) ).

% insert_compr
tff(fact_5427_insert__compr,axiom,
    ! [A2: nat,B: set_nat] : insert_nat(A2,B) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_bu(A2),B)) ).

% insert_compr
tff(fact_5428_insert__compr,axiom,
    ! [A2: int,B: set_int] : insert_int(A2,B) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_bv(A2),B)) ).

% insert_compr
tff(fact_5429_insert__compr,axiom,
    ! [A2: complex,B: set_complex] : insert_complex(A2,B) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_bw(A2),B)) ).

% insert_compr
tff(fact_5430_insert__compr,axiom,
    ! [A2: product_prod_int_int,B: set_Pr958786334691620121nt_int] : insert5033312907999012233nt_int(A2,B) = collec213857154873943460nt_int(aa_set2224860389584585407_int_o(aTP_Lamm_bx(A2),B)) ).

% insert_compr
tff(fact_5431_insert__Collect,axiom,
    ! [A2: vEBT_VEBT,P: fun_VEBT_VEBT_o] : insert_VEBT_VEBT(A2,collect_VEBT_VEBT(P)) = collect_VEBT_VEBT(aa_fun8193769468198689164VEBT_o(aTP_Lamm_by(A2),P)) ).

% insert_Collect
tff(fact_5432_insert__Collect,axiom,
    ! [A2: real,P: fun_real_o] : insert_real(A2,collect_real(P)) = collect_real(aa_fun5868358341814070780real_o(aTP_Lamm_bz(A2),P)) ).

% insert_Collect
tff(fact_5433_insert__Collect,axiom,
    ! [A2: $o,P: fun_o_o] : insert_o((A2),collect_o(P)) = collect_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ca((A2)),P)) ).

% insert_Collect
tff(fact_5434_insert__Collect,axiom,
    ! [A2: nat,P: fun_nat_o] : insert_nat(A2,collect_nat(P)) = collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_cb(A2),P)) ).

% insert_Collect
tff(fact_5435_insert__Collect,axiom,
    ! [A2: int,P: fun_int_o] : insert_int(A2,collect_int(P)) = collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_cc(A2),P)) ).

% insert_Collect
tff(fact_5436_insert__Collect,axiom,
    ! [A2: complex,P: fun_complex_o] : insert_complex(A2,collect_complex(P)) = collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_cd(A2),P)) ).

% insert_Collect
tff(fact_5437_insert__Collect,axiom,
    ! [A2: product_prod_int_int,P: fun_Pr3238374608701176162_int_o] : insert5033312907999012233nt_int(A2,collec213857154873943460nt_int(P)) = collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_ce(A2),P)) ).

% insert_Collect
tff(fact_5438_Set_Oempty__def,axiom,
    bot_bot_set_complex = collect_complex(aTP_Lamm_cf) ).

% Set.empty_def
tff(fact_5439_Set_Oempty__def,axiom,
    bot_bo1796632182523588997nt_int = collec213857154873943460nt_int(aTP_Lamm_cg) ).

% Set.empty_def
tff(fact_5440_Set_Oempty__def,axiom,
    bot_bot_set_real = collect_real(aTP_Lamm_ch) ).

% Set.empty_def
tff(fact_5441_Set_Oempty__def,axiom,
    bot_bot_set_o = collect_o(aTP_Lamm_ci) ).

% Set.empty_def
tff(fact_5442_Set_Oempty__def,axiom,
    bot_bot_set_nat = collect_nat(aTP_Lamm_cj) ).

% Set.empty_def
tff(fact_5443_Set_Oempty__def,axiom,
    bot_bot_set_int = collect_int(aTP_Lamm_ck) ).

% Set.empty_def
tff(fact_5444_lambda__one,axiom,
    aTP_Lamm_cl = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_5445_lambda__one,axiom,
    aTP_Lamm_cm = times_times_rat(one_one_rat) ).

% lambda_one
tff(fact_5446_lambda__one,axiom,
    aTP_Lamm_cn = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_5447_lambda__one,axiom,
    aTP_Lamm_co = times_times_int(one_one_int) ).

% lambda_one
tff(fact_5448_lambda__one,axiom,
    aTP_Lamm_cp = times_times_assn(one_one_assn) ).

% lambda_one
tff(fact_5449_lambda__zero,axiom,
    aTP_Lamm_cq = times_times_complex(zero_zero_complex) ).

% lambda_zero
tff(fact_5450_lambda__zero,axiom,
    aTP_Lamm_cr = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_5451_lambda__zero,axiom,
    aTP_Lamm_cs = times_times_rat(zero_zero_rat) ).

% lambda_zero
tff(fact_5452_lambda__zero,axiom,
    aTP_Lamm_ct = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_5453_lambda__zero,axiom,
    aTP_Lamm_cu = times_times_int(zero_zero_int) ).

% lambda_zero
tff(fact_5454_mult__commute__abs,axiom,
    ! [C2: real] : aa_rea7340605660300375871l_real(aTP_Lamm_cv,C2) = aa_rea7340605660300375871l_real(times_times_real,C2) ).

% mult_commute_abs
tff(fact_5455_mult__commute__abs,axiom,
    ! [C2: rat] : aTP_Lamm_cw(C2) = times_times_rat(C2) ).

% mult_commute_abs
tff(fact_5456_mult__commute__abs,axiom,
    ! [C2: nat] : aa_nat_fun_nat_nat(aTP_Lamm_cx,C2) = aa_nat_fun_nat_nat(times_times_nat,C2) ).

% mult_commute_abs
tff(fact_5457_mult__commute__abs,axiom,
    ! [C2: int] : aTP_Lamm_cy(C2) = times_times_int(C2) ).

% mult_commute_abs
tff(fact_5458_mult__commute__abs,axiom,
    ! [C2: assn] : aTP_Lamm_cz(C2) = times_times_assn(C2) ).

% mult_commute_abs
tff(fact_5459_frame__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_o,Q: fun_o_assn,R: assn] :
      ( hoare_hoare_triple_o(P,C2,Q)
     => hoare_hoare_triple_o(aa_assn_assn(times_times_assn(P),R),C2,aa_assn_fun_o_assn(aTP_Lamm_da(Q),R)) ) ).

% frame_rule
tff(fact_5460_frame__rule,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,R: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C2,Q)
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(P),R),C2,aa_ass4389913640333755968i_assn(aTP_Lamm_db(Q),R)) ) ).

% frame_rule
tff(fact_5461_frame__rule,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,R: assn] :
      ( hoare_7629718768684598413on_nat(P,C2,Q)
     => hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(P),R),C2,aa_ass4112278883933122603t_assn(aTP_Lamm_dc(Q),R)) ) ).

% frame_rule
tff(fact_5462_frame__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,R: assn] :
      ( hoare_3067605981109127869le_nat(P,C2,Q)
     => hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(P),R),C2,aa_assn_fun_nat_assn(aTP_Lamm_dd(Q),R)) ) ).

% frame_rule
tff(fact_5463_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C2: heap_Time_Heap_o,Q: fun_o_assn] :
      ( entails(P,P2)
     => ( hoare_hoare_triple_o(P2,C2,Q)
       => hoare_hoare_triple_o(P,C2,Q) ) ) ).

% cons_pre_rule
tff(fact_5464_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn] :
      ( entails(P,P2)
     => ( hoare_1429296392585015714_VEBTi(P2,C2,Q)
       => hoare_1429296392585015714_VEBTi(P,C2,Q) ) ) ).

% cons_pre_rule
tff(fact_5465_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn] :
      ( entails(P,P2)
     => ( hoare_7629718768684598413on_nat(P2,C2,Q)
       => hoare_7629718768684598413on_nat(P,C2,Q) ) ) ).

% cons_pre_rule
tff(fact_5466_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn] :
      ( entails(P,P2)
     => ( hoare_3067605981109127869le_nat(P2,C2,Q)
       => hoare_3067605981109127869le_nat(P,C2,Q) ) ) ).

% cons_pre_rule
tff(fact_5467_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect_nat(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta)) ).

% set_vebt_def
tff(fact_5468_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit0,Nb)) = $let(
        m: complex,
        m:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(m),m) ) ).

% numeral_code(2)
tff(fact_5469_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit0,Nb)) = $let(
        m: real,
        m:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m) ) ).

% numeral_code(2)
tff(fact_5470_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = $let(
        m: rat,
        m:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(m),m) ) ).

% numeral_code(2)
tff(fact_5471_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m) ) ).

% numeral_code(2)
tff(fact_5472_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(plus_plus_int(m),m) ) ).

% numeral_code(2)
tff(fact_5473_nat__less__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_less_as_int
tff(fact_5474_nat__leq__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_5475_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      numera6690914467698888265omplex(aa_num_num(bit1,Nb)) = $let(
        m: complex,
        m:= numera6690914467698888265omplex(Nb),
        aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(m),m)),one_one_complex) ) ).

% numeral_code(3)
tff(fact_5476_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_real(numeral_numeral_real,aa_num_num(bit1,Nb)) = $let(
        m: real,
        m:= aa_num_real(numeral_numeral_real,Nb),
        aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,m),m)),one_one_real) ) ).

% numeral_code(3)
tff(fact_5477_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = $let(
        m: rat,
        m:= aa_num_rat(numeral_numeral_rat,Nb),
        aa_rat_rat(plus_plus_rat(aa_rat_rat(plus_plus_rat(m),m)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_5478_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb)) = $let(
        m: nat,
        m:= aa_num_nat(numeral_numeral_nat,Nb),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,m),m)),one_one_nat) ) ).

% numeral_code(3)
tff(fact_5479_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m: int,
        m:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(m),m)),one_one_int) ) ).

% numeral_code(3)
tff(fact_5480_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(w),w) ) ).

% power_numeral_even
tff(fact_5481_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] :
      power_8256067586552552935nteger(Z,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: code_integer,
        w:= power_8256067586552552935nteger(Z,aa_num_nat(numeral_numeral_nat,W)),
        times_3573771949741848930nteger(w,w) ) ).

% power_numeral_even
tff(fact_5482_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,w),w) ) ).

% power_numeral_even
tff(fact_5483_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(w),w) ) ).

% power_numeral_even
tff(fact_5484_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,w),w) ) ).

% power_numeral_even
tff(fact_5485_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(times_times_int(w),w) ) ).

% power_numeral_even
tff(fact_5486_power__numeral__even,axiom,
    ! [Z: assn,W: num] :
      aa_nat_assn(power_power_assn(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: assn,
        w:= aa_nat_assn(power_power_assn(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_assn_assn(times_times_assn(w),w) ) ).

% power_numeral_even
tff(fact_5487_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_5488_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] :
      power_8256067586552552935nteger(Z,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: code_integer,
        w:= power_8256067586552552935nteger(Z,aa_num_nat(numeral_numeral_nat,W)),
        times_3573771949741848930nteger(times_3573771949741848930nteger(Z,w),w) ) ).

% power_numeral_odd
tff(fact_5489_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: real,
        w:= aa_nat_real(aa_real_fun_nat_real(power_power_real,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_5490_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(times_times_rat(aa_rat_rat(times_times_rat(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_5491_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: nat,
        w:= aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),w)),w) ) ).

% power_numeral_odd
tff(fact_5492_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(times_times_int(aa_int_int(times_times_int(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_5493_power__numeral__odd,axiom,
    ! [Z: assn,W: num] :
      aa_nat_assn(power_power_assn(Z),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: assn,
        w:= aa_nat_assn(power_power_assn(Z),aa_num_nat(numeral_numeral_nat,W)),
        aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(Z),w)),w) ) ).

% power_numeral_odd
tff(fact_5494_rule__at__index,axiom,
    ! [P: assn,A: fun_VE1571388752190420990T_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: fun_o_assn,F4: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L1279224858307276611T_VEBT(A,Xs,Xsi)),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F),C2,Q2)
         => ( ! [R2: $o] : entails(aa_o_assn(Q2,(R2)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_VEB115603943081001463T_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3204528365124325536T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_o_assn(F4,(R2))))
           => hoare_hoare_triple_o(P,C2,aa_fun4536517337817438332o_assn(aa_lis1936063806678856193o_assn(aa_lis5927258619744804946o_assn(aTP_Lamm_de(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5495_rule__at__index,axiom,
    ! [P: assn,A: fun_VE4931415397359990460t_assn,Xs: list_VEBT_VEBT,Xsi: list_nat,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: fun_o_assn,F4: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L8296926524756676353BT_nat(A,Xs,Xsi)),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Xs))
       => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F),C2,Q2)
         => ( ! [R2: $o] : entails(aa_o_assn(Q2,(R2)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_VEB8640158415945897581t_assn(A,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L8650695023172932196BT_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_o_assn(F4,(R2))))
           => hoare_hoare_triple_o(P,C2,aa_fun4536517337817438332o_assn(aa_lis3456267157703431713o_assn(aa_lis1555086301578635742o_assn(aTP_Lamm_df(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5496_rule__at__index,axiom,
    ! [P: assn,A: fun_re8050108611700907676T_assn,Xs: list_real,Xsi: list_VEBT_VEBT,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: fun_o_assn,F4: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L4595930785310033027T_VEBT(A,Xs,Xsi)),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F),C2,Q2)
         => ( ! [R2: $o] : entails(aa_o_assn(Q2,(R2)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_rea1965931645776870533T_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L3095048238742455910T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_o_assn(F4,(R2))))
           => hoare_hoare_triple_o(P,C2,aa_fun4536517337817438332o_assn(aa_lis1936063806678856193o_assn(aa_lis9036567463675750230o_assn(aTP_Lamm_dg(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5497_rule__at__index,axiom,
    ! [P: assn,A: fun_re560840401387337987i_assn,Xs: list_real,Xsi: list_VEBT_VEBTi,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: fun_o_assn,F4: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L9060850011106065574_VEBTi(A,Xs),Xsi)),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F),C2,Q2)
         => ( ! [R2: $o] : entails(aa_o_assn(Q2,(R2)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_rea5453075730833149764i_assn(A,aa_nat_real(nth_real(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7851252805511451907_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),aa_o_assn(F4,(R2))))
           => hoare_hoare_triple_o(P,C2,aa_fun4536517337817438332o_assn(aa_lis522499054687323654o_assn(aa_lis3287929508098246591o_assn(aTP_Lamm_dh(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5498_rule__at__index,axiom,
    ! [P: assn,A: fun_re574865750389657758t_assn,Xs: list_real,Xsi: list_nat,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: fun_o_assn,F4: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L1446010312343316929al_nat(A,Xs,Xsi)),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_real(Xs))
       => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F),C2,Q2)
         => ( ! [R2: $o] : entails(aa_o_assn(Q2,(R2)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_real_fun_nat_assn(A,aa_nat_real(nth_real(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L234762979517870878al_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_real(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_o_assn(F4,(R2))))
           => hoare_hoare_triple_o(P,C2,aa_fun4536517337817438332o_assn(aa_lis3456267157703431713o_assn(aa_lis7556923739995607514o_assn(aTP_Lamm_di(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5499_rule__at__index,axiom,
    ! [P: assn,A: fun_o_4353168661238401050T_assn,Xs: list_o,Xsi: list_VEBT_VEBT,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: fun_o_assn,F4: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L1750719106661372127T_VEBT(A,Xs,Xsi)),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L1319876754960170684T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_o(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F),C2,Q2)
         => ( ! [R2: $o] : entails(aa_o_assn(Q2,(R2)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_o_f2789175432390837779T_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L1319876754960170684T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_o(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_o_assn(F4,(R2))))
           => hoare_hoare_triple_o(P,C2,aa_fun4536517337817438332o_assn(aa_lis1936063806678856193o_assn(aa_lis6424289830231647982o_assn(aTP_Lamm_dj(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5500_rule__at__index,axiom,
    ! [P: assn,A: fun_o_9029690514724572677i_assn,Xs: list_o,Xsi: list_VEBT_VEBTi,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: fun_o_assn,F4: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L3704169666673096010_VEBTi(A,Xs),Xsi)),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A,aa_nat_o(nth_o(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_o(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F),C2,Q2)
         => ( ! [R2: $o] : entails(aa_o_assn(Q2,(R2)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_o_f7290096833689243702i_assn(A,aa_nat_o(nth_o(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L6286945158656146733_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_o(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),aa_o_assn(F4,(R2))))
           => hoare_hoare_triple_o(P,C2,aa_fun4536517337817438332o_assn(aa_lis522499054687323654o_assn(aa_lis958962040488458791o_assn(aTP_Lamm_dk(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5501_rule__at__index,axiom,
    ! [P: assn,A: fun_o_fun_nat_assn,Xs: list_o,Xsi: list_nat,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: fun_o_assn,F4: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L4785011123346445925_o_nat(A,Xs,Xsi)),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
       => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_o_fun_nat_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2281750874075065672_o_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_o(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F),C2,Q2)
         => ( ! [R2: $o] : entails(aa_o_assn(Q2,(R2)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_nat_assn(aa_o_fun_nat_assn(A,aa_nat_o(nth_o(Xs),I)),aa_nat_nat(nth_nat(Xsi),I))),vEBT_L2281750874075065672_o_nat(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_o(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_o_assn(F4,(R2))))
           => hoare_hoare_triple_o(P,C2,aa_fun4536517337817438332o_assn(aa_lis3456267157703431713o_assn(aa_lis2327745721547492418o_assn(aTP_Lamm_dl(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5502_rule__at__index,axiom,
    ! [P: assn,A: fun_na4489259242115493880T_assn,Xs: list_nat,Xsi: list_VEBT_VEBT,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: fun_o_assn,F4: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(vEBT_L8158188754432654943T_VEBT(A,Xs,Xsi)),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L8511957252848910786T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_nat(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),F),C2,Q2)
         => ( ! [R2: $o] : entails(aa_o_assn(Q2,(R2)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBT_assn(aa_nat6835651039071750625T_assn(A,aa_nat_nat(nth_nat(Xs),I)),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xsi),I))),vEBT_L8511957252848910786T_VEBT(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_nat(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs,Xsi))),aa_o_assn(F4,(R2))))
           => hoare_hoare_triple_o(P,C2,aa_fun4536517337817438332o_assn(aa_lis1936063806678856193o_assn(aa_lis765023829625672754o_assn(aTP_Lamm_dm(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5503_rule__at__index,axiom,
    ! [P: assn,A: fun_na6467670216208870567i_assn,Xs: list_nat,Xsi: list_VEBT_VEBTi,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: fun_o_assn,F4: fun_o_assn] :
      ( entails(P,aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L4387162340545308618_VEBTi(A,Xs),Xsi)),F))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
       => ( hoare_hoare_triple_o(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A,aa_nat_nat(nth_nat(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_nat(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),F),C2,Q2)
         => ( ! [R2: $o] : entails(aa_o_assn(Q2,(R2)),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_nat8143843370389587688i_assn(A,aa_nat_nat(nth_nat(Xs),I)),nth_VEBT_VEBTi(Xsi,I))),aa_lis2508312872377092907i_assn(vEBT_L7489483478785760935_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_size_list_nat(Xs)),insert_nat(I,bot_bot_set_nat)),A,Xs),Xsi))),aa_o_assn(F4,(R2))))
           => hoare_hoare_triple_o(P,C2,aa_fun4536517337817438332o_assn(aa_lis522499054687323654o_assn(aa_lis4461610605147045859o_assn(aTP_Lamm_dn(A),Xs),Xsi),F4)) ) ) ) ) ).

% rule_at_index
tff(fact_5504_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc)
    <=> $ite(
          Xc = Mia,
          $true,
          $ite(
            Xc = Maa,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xc),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_5505_vebt__member_Oelims_I2_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xc),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_5506_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
        $ite(
          Xc = Mia,
          one_one_nat,
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
            $ite(
              Xc = Maa,
              one_one_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia),
                  one_one_nat,
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                    $ite(
                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xc),
                      one_one_nat,
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                        $let(
                          h: nat,
                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_5507_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia),Mia,Xc),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xc = Mia )
                | ( Xc = Maa ) ) ),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia),Xc,Mia)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),list_u1324408373059187874T_VEBT(TreeLista,h,aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summarya),h),Summarya)),
            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_5508_vebt__member_Oelims_I1_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xc),Xaa)
      <=> (Ya) )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => ( (Ya)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
           => (Ya) )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz)
             => (Ya) )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)
               => (Ya) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)
                   => ( (Ya)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_5509_vebt__member_Oelims_I3_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xc),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
         => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz)
           => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_5510_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(Xc),Xaa) = Ya )
     => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
           => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz)
             => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)
               => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)
                   => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $ite(
                            Xaa = Mi,
                            one_one_nat,
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Ma,
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                        one_one_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                                          $let(
                                            h: nat,
                                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_5511_vebt__insert_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(Xc),Xaa) = Ya )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => ( Ya != $ite(
                  Xaa = zero_zero_nat,
                  aa_o_VEBT_VEBT(vEBT_Leaf($true),(B3)),
                  $ite(Xaa = one_one_nat,aa_o_VEBT_VEBT(vEBT_Leaf((A3)),$true),aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
             => ( Ya != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
               => ( Ya != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) ) )
           => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) )
                 => ( Ya != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                   => ( Ya != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList2,h,aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary),h),Summary)),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_5512_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xc) ),
        one_one_nat,
        $ite(
          ( ( Xc = Mia )
          & ( Xc = Maa ) ),
          one_one_nat,
          $let(
            xn: nat,
            xn:= 
              $ite(Xc = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(vEBT_vebt_mint(Summarya)))))),Xc),
            $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),l)),
                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),l)),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summarya),h),one_one_nat)),
                  one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_5513_vebt__delete_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xc) ),
        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya),
        $ite(
          ( ( Xc = Mia )
          & ( Xc = Maa ) ),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya),
          $let(
            xn: nat,
            xn:= 
              $ite(Xc = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(vEBT_vebt_mint(Summarya)))))),Xc),
            $let(
              minn: nat,
              minn:= 
                $ite(Xc = Mia,xn,Mia),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    newnode2: vEBT_VEBT,
                    newnode2:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                    $let(
                      newlist: list_VEBT_VEBT,
                      newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode2),
                      $ite(
                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode2),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summarya),h),
                          aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                  $ite(
                                    xn = Maa,
                                    $let(
                                      maxs: option_nat,
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                    Maa ))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),newlist),sn) ),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h)))),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),newlist),Summarya) ) ) ),
                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_5514_vebt__delete_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(Xc),Xaa) = Ya )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Ya != aa_o_VEBT_VEBT(vEBT_Leaf($false),(B3)) ) ) )
       => ( ! [A3: $o] :
              ( ? [B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Ya != aa_o_VEBT_VEBT(vEBT_Leaf((A3)),$false) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
               => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
                 => ( Ya != aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary) )
                 => ( Ya != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary) ) )
             => ( ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2),Smry2) )
                   => ( Ya != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2),Smry2) ) )
               => ( ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                     => ( Ya != aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Ya != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),
                              $ite(
                                ( ( Xaa = Mi )
                                & ( Xaa = Ma ) ),
                                aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi,xn,Mi),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          newnode2: vEBT_VEBT,
                                          newnode2:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                          $let(
                                            newlist: list_VEBT_VEBT,
                                            newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode2),
                                            $ite(
                                              aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode2),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),h),
                                                aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                        $ite(
                                                          xn = Ma,
                                                          $let(
                                                            maxs: option_nat,
                                                            maxs:= vEBT_vebt_maxt(sn),
                                                            $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                                          Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),sn) ),
                                              aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                      $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),Summary) ) ) ),
                                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_5515_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Xc),Xaa) = Ya )
     => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( ( Xaa = zero_zero_nat )
           => ( Ya != one_one_nat ) ) )
       => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Ya != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
             => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
               => ( Ya != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary)
               => ( Ya != one_one_nat ) )
             => ( ( ? [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList2),Summary)
                 => ( Ya != one_one_nat ) )
               => ( ( ? [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary)
                   => ( Ya != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Ya != $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                              | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                              one_one_nat,
                              $ite(
                                ( ( Xaa = Mi )
                                & ( Xaa = Ma ) ),
                                one_one_nat,
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                          $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summary),h),one_one_nat)),
                                        one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_5516__C7_OIH_C_I2_J,axiom,
    ! [Xaa: nat,Xba: nat,Xc: nat,Xd: nat,Xe: vEBT_VEBT,Xf: list_VEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi] :
      ( ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,xa),mi)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),xa) )
     => ( ~ ( ( xa = mi )
            & ( xa = ma ) )
       => ( ( Xaa = $ite(xa = mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))),xa) )
         => ( ( Xba = $ite(xa = mi,Xaa,mi) )
           => ( ( Xc = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
             => ( ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),size_s6755466524823107622T_VEBT(treeList))
                 => ( ( Xe = aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),Xd)),Xc) )
                   => ( ( Xf = list_u1324408373059187874T_VEBT(treeList,Xd,Xe) )
                     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xe)
                       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,summary),Nb)
                         => hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,summary),Tib),vEBT_V1365221501068881998eletei(summary,Tib,Xd),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(summary),Xd))) ) ) ) ) ) ) ) ) ) ) ) ).

% "7.IH"(2)
tff(fact_5517__C7_OIH_C_I1_J,axiom,
    ! [Xaa: nat,Xba: nat,Xc: nat,Xd: nat,Nb: nat,Tib: vEBT_VEBTi] :
      ( ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,xa),mi)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),xa) )
     => ( ~ ( ( xa = mi )
            & ( xa = ma ) )
       => ( ( Xaa = $ite(xa = mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))),xa) )
         => ( ( Xba = $ite(xa = mi,Xaa,mi) )
           => ( ( Xc = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
             => ( ( Xd = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xd),size_s6755466524823107622T_VEBT(treeList))
                 => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),Xd)),Nb)
                   => hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),Xd)),Tib),vEBT_V1365221501068881998eletei(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),Xd),Tib,Xc),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),Xd)),Xc))) ) ) ) ) ) ) ) ) ).

% "7.IH"(1)
tff(fact_5518_minNulli__rule,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_hoare_triple_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_VEBT_minNulli(Tib),aa_VEB6687300313434771872o_assn(aTP_Lamm_do(Ta),Tib)) ).

% minNulli_rule
tff(fact_5519_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(Xc),Xaa) = Ya )
     => ( ( ? [Uu2: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(B3))
         => ( ( Xaa = zero_zero_nat )
           => ( Ya != one_one_nat ) ) )
       => ( ( ? [Uv: $o,Uw2: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uv)),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Ya != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)
             => ( Ya != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd)
               => ( Ya != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Ya != one_one_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( Ya != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            one_one_nat,
                            $let(
                              l: nat,
                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summary),h)),one_one_nat) ) ),
                                  one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_5520_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(Xc),Xaa) = Ya )
     => ( ( ? [Uu2: $o,Uv: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv))
         => ( ( Xaa = zero_zero_nat )
           => ( Ya != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw2: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(Uw2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Ya != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
             => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
               => ( Ya != one_one_nat ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va)
               => ( Ya != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd),Ve)
                 => ( Ya != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Ya != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Ya != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              one_one_nat,
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summary),h)),one_one_nat) ) ),
                                    one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_5521_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(Xc),Xaa) = Ya )
     => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3)
           => ( Ya != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3)
             => ( Ya != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary)
               => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                   => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
                          $let(
                            xn: nat,
                            xn:= 
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                            $let(
                              h: nat,
                              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(
                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                & ~ ( ( Xaa = Mi )
                                    | ( Xaa = Ma ) ) ),
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),
                                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary),h),one_one_nat)),
                                one_one_nat ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_5522_minNull__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Ta)),one_one_nat) ).

% minNull_bound
tff(fact_5523_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Vaa: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_5524_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] : aa_nat_nat(vEBT_T_p_r_e_d2(aa_o_VEBT_VEBT(vEBT_Leaf((Uu)),(Uv2))),zero_zero_nat) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_5525_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw: $o,Nb: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(aa_o_VEBT_VEBT(vEBT_Leaf((Uv2)),(Uw))),aa_nat_nat(suc,Nb)) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_5526_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz2: list_VEBT_VEBT,Vaa: vEBT_VEBT,Vb2: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy2,Uz2),Vaa)),Vb2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_5527_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] : aa_nat_nat(vEBT_T_s_u_c_c2(aa_o_VEBT_VEBT(vEBT_Leaf((Uu)),(B2))),zero_zero_nat) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_5528_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Vaa: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy2),Uz2)),Vaa) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_5529_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Vaa: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz2),Vaa,Vb2),Vc2)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_5530_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_5531_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] : aa_nat_nat(vEBT_T_p_r_e_d2(aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(Uw))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_5532_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vd2),Ve2)),Vf2) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_5533_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Vc2),Vd2)),Ve2) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_5534_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] : aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vh),Vi)),Vj) = one_one_nat ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_5535_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] : aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vg),Vh)),Vi) = one_one_nat ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_5536_pred__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Ta),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).

% pred_bound_height'
tff(fact_5537_succ_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Ta),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).

% succ'_bound_height
tff(fact_5538_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xc) = Ya )
     => ( ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) )
         => ( Ya != one_one_nat ) )
       => ( ( ? [Uv: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf($true),(Uv))
           => ( Ya != one_one_nat ) )
         => ( ( ? [Uu2: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),$true)
             => ( Ya != one_one_nat ) )
           => ( ( ? [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux),Uy)
               => ( Ya != one_one_nat ) )
             => ~ ( ? [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc)
                 => ( Ya != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_5539_pred__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_p_r_e_d2(Ta),Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_5540_succ__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ( U = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(vEBT_T_s_u_c_c2(Ta),Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_5541_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
        $let(
          xn: nat,
          xn:= 
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia),Mia,Xc),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
              & ~ ( ( Xc = Mia )
                  | ( Xc = Maa ) ) ),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)))),
                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summarya),h),one_one_nat)),
              one_one_nat ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_5542_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xc),
        one_one_nat,
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),l)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summarya),h)),one_one_nat) ) ),
              one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_5543_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_nat(vEBT_T_s_u_c_c2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia),
        one_one_nat,
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),l)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summarya),h)),one_one_nat) ) ),
              one_one_nat ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_5544_minNrulli__ruleT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_VEBT_minNulli(Tib),aa_VEB6687300313434771872o_assn(aTP_Lamm_do(Ta),Tib)),one_one_nat) ).

% minNrulli_ruleT
tff(fact_5545__092_060open_062_092_060And_062x22_Ax21_O_Ati_A_061_ALeafi_Ax21_Ax22_A_092_060Longrightarrow_062_A_060vebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_062_Avebt__deletei_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_Ax_A_060vebt__assn__raw_A_Ivebt__delete_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ax_J_062_092_060close_062,axiom,
    ! [X21: $o,X222: $o] :
      ( ( tia = aa_o_VEBT_VEBTi(vEBT_Leafi((X21)),(X222)) )
     => hoare_1429296392585015714_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary)),tia),vEBT_V1365221501068881998eletei(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary),tia,xa),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),aa_nat_nat(suc,aa_nat_nat(suc,va)),treeList),summary)),xa))) ) ).

% \<open>\<And>x22 x21. ti = Leafi x21 x22 \<Longrightarrow> <vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti> vebt_deletei' (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti x <vebt_assn_raw (vebt_delete (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) x)>\<close>
tff(fact_5546_builupi_Hcorr,axiom,
    ! [Nb: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb))) ).

% builupi'corr
tff(fact_5547_builupicorr,axiom,
    ! [Nb: nat] : hoare_1429296392585015714_VEBTi(one_one_assn,vEBT_vebt_buildupi(Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb))) ).

% builupicorr
tff(fact_5548_vebt__mintilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts2))
     => hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts2),Tsi),vEBT_vebt_minti(nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_dp(I),Ts2),Tsi)) ) ).

% vebt_mintilist
tff(fact_5549_vebt__maxtilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(Ts2))
     => hoare_7629718768684598413on_nat(aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Ts2),Tsi),vEBT_vebt_maxti(nth_VEBT_VEBTi(Tsi,I)),aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_dq(I),Ts2),Tsi)) ) ).

% vebt_maxtilist
tff(fact_5550_vebt__minti__h,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_minti(Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_dr(Ta),Tib)) ).

% vebt_minti_h
tff(fact_5551_vebt__maxti__h,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_7629718768684598413on_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_maxti(Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_ds(Ta),Tib)) ).

% vebt_maxti_h
tff(fact_5552_htt__vebt__memberi,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xc: nat] : aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_memberi(Tib,Xc),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_ao(Ta),Tib),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)))) ).

% htt_vebt_memberi
tff(fact_5553_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,aa_o_VEBT_VEBTi(vEBT_Leafi((X21)),(X222))) = zero_zero_nat ).

% VEBTi.size(4)
tff(fact_5554_VEBTi_Odistinct_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12: nat,X13a: array_VEBT_VEBTi,X14a: vEBT_VEBTi,X21: $o,X222: $o] : aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11a,X12,X13a),X14a) != aa_o_VEBT_VEBTi(vEBT_Leafi((X21)),(X222)) ).

% VEBTi.distinct(1)
tff(fact_5555_VEBTi_Oexhaust,axiom,
    ! [Ya: vEBT_VEBTi] :
      ( ! [X11: option4927543243414619207at_nat,X122: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] : Ya != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11,X122,X13),X14)
     => ~ ! [X212: $o,X223: $o] : Ya != aa_o_VEBT_VEBTi(vEBT_Leafi((X212)),(X223)) ) ).

% VEBTi.exhaust
tff(fact_5556_vebt__assn__raw_Ocases,axiom,
    ! [Xc: produc3625547720036274456_VEBTi] :
      ( ! [A3: $o,B3: $o,Ai: $o,Bi: $o] : Xc != produc6084888613844515218_VEBTi(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_o_VEBT_VEBTi(vEBT_Leafi((Ai)),(Bi)))
     => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary: vEBT_VEBT,Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] : Xc != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi))
       => ( ! [V2: option4927543243414619207at_nat,Va2: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] : Xc != produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3),aa_o_VEBT_VEBTi(vEBT_Leafi((Vd3)),(Ve3)))
         => ~ ! [Vd3: $o,Ve3: $o,V2: option4927543243414619207at_nat,Va2: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : Xc != produc6084888613844515218_VEBTi(aa_o_VEBT_VEBT(vEBT_Leaf((Vd3)),(Ve3)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3)) ) ) ) ).

% vebt_assn_raw.cases
tff(fact_5557_VEBT__internal_OminNulli_Ocases,axiom,
    ! [Xc: vEBT_VEBTi] :
      ( ( Xc != aa_o_VEBT_VEBTi(vEBT_Leafi($false),$false) )
     => ( ! [Uv: $o] : Xc != aa_o_VEBT_VEBTi(vEBT_Leafi($true),(Uv))
       => ( ! [Uu2: $o] : Xc != aa_o_VEBT_VEBTi(vEBT_Leafi((Uu2)),$true)
         => ( ! [Uw2: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] : Xc != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux),Uy)
           => ~ ! [Uz: product_prod_nat_nat,Va: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] : Xc != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc) ) ) ) ) ).

% VEBT_internal.minNulli.cases
tff(fact_5558_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Ai2: $o,Bi2: $o] :
      aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),aa_o_VEBT_VEBTi(vEBT_Leafi((Ai2)),(Bi2))) = aa_o_assn(pure_assn,
        ( ( (Ai2) = (A2) )
        & ( (Bi2) = (B2) ) )) ).

% vebt_assn_raw.simps(1)
tff(fact_5559_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option4927543243414619207at_nat,Vaa: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,Vd2: $o,Ve2: $o] : aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(V,Vaa,Vb2),Vc2)),aa_o_VEBT_VEBTi(vEBT_Leafi((Vd2)),(Ve2))) = bot_bot_assn ).

% vebt_assn_raw.simps(3)
tff(fact_5560_vebt__minti_Ocases,axiom,
    ! [Xc: vEBT_VEBTi] :
      ( ! [A3: $o,B3: $o] : Xc != aa_o_VEBT_VEBTi(vEBT_Leafi((A3)),(B3))
     => ( ! [Uu2: nat,Uv: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : Xc != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
       => ~ ! [Mi: nat,Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : Xc != aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz) ) ) ).

% vebt_minti.cases
tff(fact_5561_vebt__buildupi__rule,axiom,
    ! [Nb: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_o_assn(pure_assn,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)),vEBT_vebt_buildupi(Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% vebt_buildupi_rule
tff(fact_5562_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% htt_vebt_buildupi_univ
tff(fact_5563_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),U)) ) ).

% htt_vebt_buildupi'_univ
tff(fact_5564_vebt__maxti__hT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_maxti(Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_ds(Ta),Tib)),one_one_nat) ).

% vebt_maxti_hT
tff(fact_5565_vebt__minti__hT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_minti(Tib),aa_VEB7611521244314475576t_assn(aTP_Lamm_dr(Ta),Tib)),one_one_nat) ).

% vebt_minti_hT
tff(fact_5566_T__vebt__buildupi,axiom,
    ! [Nb: nat,H: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(vEBT_V739175172307565963ildupi(Nb),H)),aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) ).

% T_vebt_buildupi
tff(fact_5567_htt__vebt__buildupi_H,axiom,
    ! [Nb: nat] : aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_V739175172307565963ildupi(Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb))),aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) ).

% htt_vebt_buildupi'
tff(fact_5568_htt__vebt__buildupi,axiom,
    ! [Nb: nat] : aa_nat_o(time_htt_VEBT_VEBTi(one_one_assn,vEBT_vebt_buildupi(Nb),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb))),aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) ).

% htt_vebt_buildupi
tff(fact_5569_htt__vebt__inserti,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xc: nat] : aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_inserti(Tib,Xc),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Ta),Xc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)))) ).

% htt_vebt_inserti
tff(fact_5570_time__replicate,axiom,
    ! [Xc: heap_T8145700208782473153_VEBTi,C2: nat,Nb: nat,H: heap_e7401611519738050253t_unit] :
      ( ! [H4: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(Xc,H4)),C2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_t3534373299052942712_VEBTi(vEBT_V1859673955506687831_VEBTi(Nb,Xc),H)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C2)),Nb))) ) ).

% time_replicate
tff(fact_5571_htt__vebt__memberi__invar__vebt,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(time_htt_o(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_memberi(Tib,Xc),aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_ao(Ta),Tib),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_5572_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Xc),Xaa) = Ya )
     => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( Ya != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
           => ( Ya != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz)
             => ( Ya != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc)
               => ( Ya != one_one_nat ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary)
                   => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                          $ite(
                            Xaa = Mi,
                            zero_zero_nat,
                            $ite(
                              Xaa = Ma,
                              zero_zero_nat,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                zero_zero_nat,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  zero_zero_nat,
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa)
                                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma) ),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                                    zero_zero_nat ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_5573_TBOUND__vebt__inserti,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xc: nat] :
      aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_V3964819847710782039nserti(Ta,Tib,Xc)),
        $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))))) ).

% TBOUND_vebt_inserti
tff(fact_5574_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Xc),Xaa) = Ya )
     => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( Ya != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3)
           => ( Ya != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3)
             => ( Ya != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary)
               => ( Ya != one_one_nat ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                   => ( Ya != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary),h),one_one_nat)),
                              one_one_nat ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_5575_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          Xc = Mia,
          zero_zero_nat,
          $ite(
            Xc = Maa,
            zero_zero_nat,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia),
              zero_zero_nat,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xc),
                zero_zero_nat,
                $ite(
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xc)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Maa) ),
                  $let(
                    h: nat,
                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                  zero_zero_nat ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_5576_TBOUND__vebt__buildupi,axiom,
    ! [Nb: nat] : aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_V739175172307565963ildupi(Nb)),aa_nat_nat(vEBT_V441764108873111860ildupi,Nb)) ).

% TBOUND_vebt_buildupi
tff(fact_5577_TBOUND__minNull,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xc: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_V3964819847710782039nserti(Ta,Tib,Xc)),one_one_nat) ) ).

% TBOUND_minNull
tff(fact_5578_TBOUND__replicate,axiom,
    ! [Xc: heap_T8145700208782473153_VEBTi,C2: nat,Nb: nat] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(Xc),C2)
     => aa_nat_o(time_T8149879359713347829_VEBTi(vEBT_V1859673955506687831_VEBTi(Nb,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C2)),Nb))) ) ).

% TBOUND_replicate
tff(fact_5579_TBOUND__replicate,axiom,
    ! [Xc: heap_Time_Heap_o,C2: nat,Nb: nat] :
      ( aa_nat_o(time_TBOUND_o(Xc),C2)
     => aa_nat_o(time_TBOUND_list_o(vEBT_V2326993469660664182atei_o(Nb,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C2)),Nb))) ) ).

% TBOUND_replicate
tff(fact_5580_TBOUND__replicate,axiom,
    ! [Xc: heap_T2636463487746394924on_nat,C2: nat,Nb: nat] :
      ( aa_nat_o(time_T8353473612707095248on_nat(Xc),C2)
     => aa_nat_o(time_T3808005469503390304on_nat(vEBT_V792416675989592002on_nat(Nb,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C2)),Nb))) ) ).

% TBOUND_replicate
tff(fact_5581_TBOUND__replicate,axiom,
    ! [Xc: heap_Time_Heap_nat,C2: nat,Nb: nat] :
      ( aa_nat_o(time_TBOUND_nat(Xc),C2)
     => aa_nat_o(time_TBOUND_list_nat(vEBT_V7726092123322077554ei_nat(Nb,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),C2)),Nb))) ) ).

% TBOUND_replicate
tff(fact_5582_TBOUND__buildupi,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_vebt_buildupi(Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% TBOUND_buildupi
tff(fact_5583_nat__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_num_int(numeral_numeral_int,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_numeral
tff(fact_5584_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_5585_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa_int_nat(nat2,I) = zero_zero_nat )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int) ) ).

% nat_0_iff
tff(fact_5586_nat__le__0,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,Z) = zero_zero_nat ) ) ).

% nat_le_0
tff(fact_5587_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% zless_nat_conj
tff(fact_5588_htt__vebt__inserti__invar__vebt,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(time_htt_VEBT_VEBTi(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_inserti(Tib,Xc),aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(vEBT_vebt_insert(Ta),Xc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_5589_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z) ) ).

% zero_less_nat_eq
tff(fact_5590_diff__nat__numeral,axiom,
    ! [V: num,V3: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),aa_num_nat(numeral_numeral_nat,V3)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V3))) ).

% diff_nat_numeral
tff(fact_5591_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) = aa_int_nat(nat2,Ya) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) = Ya ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_5592_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Ya: int,Xc: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Ya) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb) )
    <=> ( Ya = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_5593_nat__ceiling__le__eq,axiom,
    ! [Xc: real,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xc))),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_nat_real(semiri5074537144036343181t_real,A2)) ) ).

% nat_ceiling_le_eq
tff(fact_5594_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Z) ) ).

% one_less_nat_eq
tff(fact_5595_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_5596_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb)),aa_int_nat(nat2,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_5597_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_5598_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_5599_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xc)),Nb)),aa_int_nat(nat2,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xc)),Nb)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_5600_htt__vebt__succi,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_succi(Tib,Xc),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_ap(Ta),Tib),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))))))) ) ).

% htt_vebt_succi
tff(fact_5601_htt__vebt__predi,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(time_htt_option_nat(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_predi(Tib,Xc),aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_aq(Ta),Tib),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))))))) ) ).

% htt_vebt_predi
tff(fact_5602_nat__numeral__as__int,axiom,
    ! [X4: num] : aa_num_nat(numeral_numeral_nat,X4) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X4)) ).

% nat_numeral_as_int
tff(fact_5603_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_5604_nat__mono,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xc)),aa_int_nat(nat2,Ya)) ) ).

% nat_mono
tff(fact_5605_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_5606_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_int_nat(nat2,Z))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_5607_nat__le__iff,axiom,
    ! [Xc: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xc)),Nb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_le_iff
tff(fact_5608_of__nat__ceiling,axiom,
    ! [R3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R3),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,R3)))) ).

% of_nat_ceiling
tff(fact_5609_of__nat__ceiling,axiom,
    ! [R3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R3),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim2889992004027027881ng_rat,R3)))) ).

% of_nat_ceiling
tff(fact_5610_nat__int__add,axiom,
    ! [A2: nat,B2: nat] : aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2) ).

% nat_int_add
tff(fact_5611_int__minus,axiom,
    ! [Nb: nat,M: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,M)))) ).

% int_minus
tff(fact_5612_real__nat__ceiling__ge,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xc)))) ).

% real_nat_ceiling_ge
tff(fact_5613_nat__plus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_5614_nat__times__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_5615_nat__minus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_minus_as_int
tff(fact_5616_nat__div__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_5617_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_5618_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa_int_nat(nat2,W) = M )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff
tff(fact_5619_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa_int_nat(nat2,W) )
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W),W = aa_nat_int(semiri1314217659103216013at_int,M),M = zero_zero_nat) ) ).

% nat_eq_iff2
tff(fact_5620_split__nat,axiom,
    ! [P: fun_nat_o,I: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,I))
    <=> ( ! [N6: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N6) )
           => aa_nat_o(P,N6) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_5621_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),W)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_5622_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
       => ( aa_int_nat(nat2,aa_int_int(plus_plus_int(Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_5623_le__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K) ) ) ).

% le_nat_iff
tff(fact_5624_Suc__as__int,axiom,
    ! [X4: nat] : aa_nat_nat(suc,X4) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,X4)),one_one_int)) ).

% Suc_as_int
tff(fact_5625_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_int_int(times_times_int(Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ).

% nat_mult_distrib
tff(fact_5626_nat__diff__distrib_H,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => ( aa_int_nat(nat2,aa_int_int(minus_minus_int(Xc),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xc)),aa_int_nat(nat2,Ya)) ) ) ) ).

% nat_diff_distrib'
tff(fact_5627_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z5)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z5),Z)
       => ( aa_int_nat(nat2,aa_int_int(minus_minus_int(Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_5628_nat__div__distrib_H,axiom,
    ! [Ya: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xc),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xc)),aa_int_nat(nat2,Ya)) ) ) ).

% nat_div_distrib'
tff(fact_5629_nat__div__distrib,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xc),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xc)),aa_int_nat(nat2,Ya)) ) ) ).

% nat_div_distrib
tff(fact_5630_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Z),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_5631_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,zero_zero_nat,Ts2),S2)),Xc) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_5632_nat__2,axiom,
    aa_int_nat(nat2,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat)) ).

% nat_2
tff(fact_5633_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Z)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(one_one_int),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_5634_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),M)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,M)) ) ) ).

% nat_less_iff
tff(fact_5635_diff__nat__eq__if,axiom,
    ! [Z: int,Z5: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Z)),aa_int_nat(nat2,Z5)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z5),zero_zero_int),
        aa_int_nat(nat2,Z),
        $let(
          d: int,
          d:= aa_int_int(minus_minus_int(Z),Z5),
          $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,d),zero_zero_int),zero_zero_nat,aa_int_nat(nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_5636_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw)),Xc) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_5637_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts2),S2)),Xc) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_5638_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista),Summarya)),Xc) = one_one_nat ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_5639_insersimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Ya: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( ~ ? [X_12: nat] : aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),X_12)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Ta),Ya)),one_one_nat) ) ) ).

% insersimp'
tff(fact_5640_insertsimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Ta),L)),one_one_nat) ) ) ).

% insertsimp'
tff(fact_5641_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),zero_zero_nat,Uy2),Uz2)),Xc) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_5642_insert_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Ta),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).

% insert'_bound_height
tff(fact_5643_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xc: nat] : aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb2),Vc2)),Xc) = one_one_nat ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_5644_member__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Ta),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta))) ) ).

% member_bound_height'
tff(fact_5645_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia),Mia,Xc),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xc = Mia )
                | ( Xc = Maa ) ) ),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summarya),h),one_one_nat)),
            one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_5646_TBOUND__vebt__memberi,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xc: nat] : aa_nat_o(time_TBOUND_o(vEBT_V854960066525838166emberi(Ta,Tib,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)))) ).

% TBOUND_vebt_memberi
tff(fact_5647_httI,axiom,
    ! [P: assn,C2: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat] :
      ( hoare_hoare_triple_o(P,C2,Q)
     => ( ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H4,As))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(C2,H4)),Ta) )
       => aa_nat_o(time_htt_o(P,C2,Q),Ta) ) ) ).

% httI
tff(fact_5648_httI,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( hoare_1429296392585015714_VEBTi(P,C2,Q)
     => ( ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H4,As))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(C2,H4)),Ta) )
       => aa_nat_o(time_htt_VEBT_VEBTi(P,C2,Q),Ta) ) ) ).

% httI
tff(fact_5649_httI,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat] :
      ( hoare_7629718768684598413on_nat(P,C2,Q)
     => ( ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H4,As))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(C2,H4)),Ta) )
       => aa_nat_o(time_htt_option_nat(P,C2,Q),Ta) ) ) ).

% httI
tff(fact_5650_httI,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat] :
      ( hoare_3067605981109127869le_nat(P,C2,Q)
     => ( ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H4,As))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(C2,H4)),Ta) )
       => aa_nat_o(time_htt_nat(P,C2,Q),Ta) ) ) ).

% httI
tff(fact_5651_htt__def,axiom,
    ! [P: assn,C2: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat] :
      ( aa_nat_o(time_htt_o(P,C2,Q),Ta)
    <=> ( hoare_hoare_triple_o(P,C2,Q)
        & ! [H3: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H3,As2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(C2,H3)),Ta) ) ) ) ).

% htt_def
tff(fact_5652_htt__def,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(P,C2,Q),Ta)
    <=> ( hoare_1429296392585015714_VEBTi(P,C2,Q)
        & ! [H3: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H3,As2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(C2,H3)),Ta) ) ) ) ).

% htt_def
tff(fact_5653_htt__def,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_option_nat(P,C2,Q),Ta)
    <=> ( hoare_7629718768684598413on_nat(P,C2,Q)
        & ! [H3: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H3,As2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(C2,H3)),Ta) ) ) ) ).

% htt_def
tff(fact_5654_htt__def,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_nat(P,C2,Q),Ta)
    <=> ( hoare_3067605981109127869le_nat(P,C2,Q)
        & ! [H3: heap_e7401611519738050253t_unit,As2: set_nat] :
            ( aa_Pro653082453696054616_nat_o(rep_assn(P),produc7507926704131184380et_nat(H3,As2))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(C2,H3)),Ta) ) ) ) ).

% htt_def
tff(fact_5655_TBOUND__vebt__succi,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xc: nat] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_VEBT_vebt_succi(Ta,Tib,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)))) ).

% TBOUND_vebt_succi
tff(fact_5656_TBOUND__vebt__predi,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xc: nat] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_VEBT_vebt_predi(Ta,Tib,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)))) ).

% TBOUND_vebt_predi
tff(fact_5657_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(Xc),Xaa) = Ya )
     => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( ( Xaa = zero_zero_nat )
           => ( Ya != one_one_nat ) ) )
       => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Ya != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
             => ( ? [N: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N))
               => ( Ya != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary)
               => ( Ya != one_one_nat ) )
             => ( ( ? [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList2),Summary)
                 => ( Ya != one_one_nat ) )
               => ( ( ? [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary)
                   => ( Ya != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                              $ite(
                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma ) ),
                                    aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                                              $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                                      $let(
                                        xn: nat,
                                        xn:= 
                                          $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                        $let(
                                          l: nat,
                                          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $let(
                                            h: nat,
                                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                            $ite(
                                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l))),
                                                $let(
                                                  newnode2: vEBT_VEBT,
                                                  newnode2:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l),
                                                  $let(
                                                    newlist: list_VEBT_VEBT,
                                                    newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode2),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode2))),
                                                      $ite(
                                                        aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode2),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_d_e_l_e_t_e(Summary),h))),
                                                          $let(
                                                            sn: vEBT_VEBT,
                                                            sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),h),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                              $ite(
                                                                xn = Ma,
                                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                                                  $let(
                                                                    maxs: option_nat,
                                                                    maxs:= vEBT_vebt_maxt(sn),
                                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                                      $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) )),
                                                                one_one_nat )) )),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                          $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h))),one_one_nat)) )) ) )),
                                              one_one_nat ) ) ) )) )) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_5658_TBOUND__vebt__maxti,axiom,
    ! [Ta: vEBT_VEBTi] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_vebt_maxti(Ta)),one_one_nat) ).

% TBOUND_vebt_maxti
tff(fact_5659_TBOUND__vebt__minti,axiom,
    ! [Ta: vEBT_VEBTi] : aa_nat_o(time_T8353473612707095248on_nat(vEBT_vebt_minti(Ta)),one_one_nat) ).

% TBOUND_vebt_minti
tff(fact_5660_TBOUND__minNulli,axiom,
    ! [Ta: vEBT_VEBTi] : aa_nat_o(time_TBOUND_o(vEBT_VEBT_minNulli(Ta)),one_one_nat) ).

% TBOUND_minNulli
tff(fact_5661_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((B2),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_5662_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_5663_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv2),Uw)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_5664_maxt__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Ta)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).

% maxt_bound
tff(fact_5665_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Ux2,Uy2),Uz2)) = one_one_nat ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_5666_mint__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Ta)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) ).

% mint_bound
tff(fact_5667_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite((A2),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_5668_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Ux2,Uy2),Uz2)) = one_one_nat ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_5669_TBOUND__mono,axiom,
    ! [C2: heap_T8145700208782473153_VEBTi,Ta: nat,T2: nat] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(C2),Ta)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),T2)
       => aa_nat_o(time_T5737551269749752165_VEBTi(C2),T2) ) ) ).

% TBOUND_mono
tff(fact_5670_TBOUND__mono,axiom,
    ! [C2: heap_Time_Heap_o,Ta: nat,T2: nat] :
      ( aa_nat_o(time_TBOUND_o(C2),Ta)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),T2)
       => aa_nat_o(time_TBOUND_o(C2),T2) ) ) ).

% TBOUND_mono
tff(fact_5671_TBOUND__mono,axiom,
    ! [C2: heap_T2636463487746394924on_nat,Ta: nat,T2: nat] :
      ( aa_nat_o(time_T8353473612707095248on_nat(C2),Ta)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),T2)
       => aa_nat_o(time_T8353473612707095248on_nat(C2),T2) ) ) ).

% TBOUND_mono
tff(fact_5672_TBOUND__mono,axiom,
    ! [C2: heap_Time_Heap_nat,Ta: nat,T2: nat] :
      ( aa_nat_o(time_TBOUND_nat(C2),Ta)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),T2)
       => aa_nat_o(time_TBOUND_nat(C2),T2) ) ) ).

% TBOUND_mono
tff(fact_5673_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xc) = Ya )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((B3),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
           => ( Ya != one_one_nat ) )
         => ~ ( ? [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)
             => ( Ya != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_5674_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xc) = Ya )
     => ( ! [A3: $o] :
            ( ? [B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
           => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                  $ite((A3),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
           => ( Ya != one_one_nat ) )
         => ~ ( ? [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)
             => ( Ya != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_5675_TBOUNDD,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,Ta: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(M),Ta)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H)),Ta) ) ).

% TBOUNDD
tff(fact_5676_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_o,Ta: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_TBOUND_o(M),Ta)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(M,H)),Ta) ) ).

% TBOUNDD
tff(fact_5677_TBOUNDD,axiom,
    ! [M: heap_T2636463487746394924on_nat,Ta: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_T8353473612707095248on_nat(M),Ta)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(M,H)),Ta) ) ).

% TBOUNDD
tff(fact_5678_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_nat,Ta: nat,H: heap_e7401611519738050253t_unit] :
      ( aa_nat_o(time_TBOUND_nat(M),Ta)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(M,H)),Ta) ) ).

% TBOUNDD
tff(fact_5679_TBOUNDI,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,Ta: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H4)),Ta)
     => aa_nat_o(time_T5737551269749752165_VEBTi(M),Ta) ) ).

% TBOUNDI
tff(fact_5680_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_o,Ta: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(M,H4)),Ta)
     => aa_nat_o(time_TBOUND_o(M),Ta) ) ).

% TBOUNDI
tff(fact_5681_TBOUNDI,axiom,
    ! [M: heap_T2636463487746394924on_nat,Ta: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(M,H4)),Ta)
     => aa_nat_o(time_T8353473612707095248on_nat(M),Ta) ) ).

% TBOUNDI
tff(fact_5682_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_nat,Ta: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(M,H4)),Ta)
     => aa_nat_o(time_TBOUND_nat(M),Ta) ) ).

% TBOUNDI
tff(fact_5683_TBOUND__def,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,Ta: nat] :
      ( aa_nat_o(time_T5737551269749752165_VEBTi(M),Ta)
    <=> ! [H3: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_VEBT_VEBTi(M,H3)),Ta) ) ).

% TBOUND_def
tff(fact_5684_TBOUND__def,axiom,
    ! [M: heap_Time_Heap_o,Ta: nat] :
      ( aa_nat_o(time_TBOUND_o(M),Ta)
    <=> ! [H3: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_o(M,H3)),Ta) ) ).

% TBOUND_def
tff(fact_5685_TBOUND__def,axiom,
    ! [M: heap_T2636463487746394924on_nat,Ta: nat] :
      ( aa_nat_o(time_T8353473612707095248on_nat(M),Ta)
    <=> ! [H3: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_option_nat(M,H3)),Ta) ) ).

% TBOUND_def
tff(fact_5686_TBOUND__def,axiom,
    ! [M: heap_Time_Heap_nat,Ta: nat] :
      ( aa_nat_o(time_TBOUND_nat(M),Ta)
    <=> ! [H3: heap_e7401611519738050253t_unit] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,time_time_nat(M,H3)),Ta) ) ).

% TBOUND_def
tff(fact_5687_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: $o,P: assn,F2: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat] :
      ( aa_nat_o(time_htt_o(aa_assn_assn(times_times_assn(aa_o_assn(pure_assn,(B2))),P),F2,Q),Ta)
    <=> ( (B2)
       => aa_nat_o(time_htt_o(P,F2,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_5688_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: $o,P: assn,F2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(aa_assn_assn(times_times_assn(aa_o_assn(pure_assn,(B2))),P),F2,Q),Ta)
    <=> ( (B2)
       => aa_nat_o(time_htt_VEBT_VEBTi(P,F2,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_5689_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: $o,P: assn,F2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_option_nat(aa_assn_assn(times_times_assn(aa_o_assn(pure_assn,(B2))),P),F2,Q),Ta)
    <=> ( (B2)
       => aa_nat_o(time_htt_option_nat(P,F2,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_5690_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: $o,P: assn,F2: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_nat(aa_assn_assn(times_times_assn(aa_o_assn(pure_assn,(B2))),P),F2,Q),Ta)
    <=> ( (B2)
       => aa_nat_o(time_htt_nat(P,F2,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_5691_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: $o,F2: heap_Time_Heap_o,Q: fun_o_assn,Ta: nat] :
      ( aa_nat_o(time_htt_o(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q),Ta)
    <=> ( (B2)
       => aa_nat_o(time_htt_o(P,F2,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_5692_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: $o,F2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q),Ta)
    <=> ( (B2)
       => aa_nat_o(time_htt_VEBT_VEBTi(P,F2,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_5693_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: $o,F2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_option_nat(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q),Ta)
    <=> ( (B2)
       => aa_nat_o(time_htt_option_nat(P,F2,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_5694_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: $o,F2: heap_Time_Heap_nat,Q: fun_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_nat(aa_assn_assn(times_times_assn(P),aa_o_assn(pure_assn,(B2))),F2,Q),Ta)
    <=> ( (B2)
       => aa_nat_o(time_htt_nat(P,F2,Q),Ta) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_5695_htt__cons__rule,axiom,
    ! [P2: assn,C2: heap_Time_Heap_o,Q2: fun_o_assn,T2: nat,P: assn,Q: fun_o_assn,Ta: nat] :
      ( aa_nat_o(time_htt_o(P2,C2,Q2),T2)
     => ( entails(P,P2)
       => ( ! [X3: $o] : entails(aa_o_assn(Q2,(X3)),aa_o_assn(Q,(X3)))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T2),Ta)
           => aa_nat_o(time_htt_o(P,C2,Q),Ta) ) ) ) ) ).

% htt_cons_rule
tff(fact_5696_htt__cons__rule,axiom,
    ! [P2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: fun_VEBT_VEBTi_assn,T2: nat,P: assn,Q: fun_VEBT_VEBTi_assn,Ta: nat] :
      ( aa_nat_o(time_htt_VEBT_VEBTi(P2,C2,Q2),T2)
     => ( entails(P,P2)
       => ( ! [X3: vEBT_VEBTi] : entails(aa_VEBT_VEBTi_assn(Q2,X3),aa_VEBT_VEBTi_assn(Q,X3))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T2),Ta)
           => aa_nat_o(time_htt_VEBT_VEBTi(P,C2,Q),Ta) ) ) ) ) ).

% htt_cons_rule
tff(fact_5697_htt__cons__rule,axiom,
    ! [P2: assn,C2: heap_T2636463487746394924on_nat,Q2: fun_option_nat_assn,T2: nat,P: assn,Q: fun_option_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_option_nat(P2,C2,Q2),T2)
     => ( entails(P,P2)
       => ( ! [X3: option_nat] : entails(aa_option_nat_assn(Q2,X3),aa_option_nat_assn(Q,X3))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T2),Ta)
           => aa_nat_o(time_htt_option_nat(P,C2,Q),Ta) ) ) ) ) ).

% htt_cons_rule
tff(fact_5698_htt__cons__rule,axiom,
    ! [P2: assn,C2: heap_Time_Heap_nat,Q2: fun_nat_assn,T2: nat,P: assn,Q: fun_nat_assn,Ta: nat] :
      ( aa_nat_o(time_htt_nat(P2,C2,Q2),T2)
     => ( entails(P,P2)
       => ( ! [X3: nat] : entails(aa_nat_assn(Q2,X3),aa_nat_assn(Q,X3))
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,T2),Ta)
           => aa_nat_o(time_htt_nat(P,C2,Q),Ta) ) ) ) ) ).

% htt_cons_rule
tff(fact_5699_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_nat(vEBT_T_p_r_e_d(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xc),
          one_one_nat,
          $let(
            l: nat,
            l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),
                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                      $ite(
                        ( ( minlow != none_nat )
                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_p_r_e_d(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),l)),
                        $let(
                          pr: option_nat,
                          pr:= vEBT_vebt_pred(Summarya,h),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summarya),h))),one_one_nat)),
                            $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(pr)))))) ) )) ),
                  one_one_nat )) ) ) )) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_5700_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_nat(vEBT_T_s_u_c_c(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
        $ite(
          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia),
          one_one_nat,
          $let(
            l: nat,
            l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $let(
              h: nat,
              h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)))),
                    $let(
                      maxlow: option_nat,
                      maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                        $ite(
                          ( ( maxlow != none_nat )
                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_s_u_c_c(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),l)),
                          $let(
                            sc: option_nat,
                            sc:= vEBT_vebt_succ(Summarya,h),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summarya),h))),one_one_nat)),
                              $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(sc)))))) ) )) )),
                  one_one_nat )) ) ) )) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_5701_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(Xc),Xaa) = Ya )
     => ( ( ? [Uu2: $o,Uv: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv))
         => ( ( Xaa = zero_zero_nat )
           => ( Ya != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw2: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(Uw2))
           => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
             => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $ite((B3),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va)
               => ( Ya != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd),Ve)
                 => ( Ya != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Ya != one_one_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                one_one_nat,
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                                            $ite(
                                              ( ( minlow != none_nat )
                                              & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_p_r_e_d(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                              $let(
                                                pr: option_nat,
                                                pr:= vEBT_vebt_pred(Summary,h),
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summary),h))),one_one_nat)),
                                                  $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(pr)))))) ) )) ),
                                        one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_5702_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(Xc),Xaa) = Ya )
     => ( ( ? [Uu2: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(B3))
         => ( ( Xaa = zero_zero_nat )
           => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) ) ) )
       => ( ( ? [Uv: $o,Uw2: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uv)),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Ya != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)
             => ( Ya != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd)
               => ( Ya != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Ya != one_one_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                              one_one_nat,
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),
                                        $let(
                                          maxlow: option_nat,
                                          maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                            $ite(
                                              ( ( maxlow != none_nat )
                                              & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_s_u_c_c(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                              $let(
                                                sc: option_nat,
                                                sc:= vEBT_vebt_succ(Summary,h),
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summary),h))),one_one_nat)),
                                                  $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(sc)))))) ) )) )),
                                      one_one_nat )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_5703_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya)),Xc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
        $ite(
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xc) ),
          one_one_nat,
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
            $ite(
              ( ( Xc = Mia )
              & ( Xc = Maa ) ),
              aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                        $ite(Xc = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summarya)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(vEBT_vebt_mint(Summarya)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                $let(
                  xn: nat,
                  xn:= 
                    $ite(Xc = Mia,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(vEBT_vebt_mint(Summarya)))))),Xc),
                  $let(
                    l: nat,
                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $let(
                      h: nat,
                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),l))),
                          $let(
                            newnode2: vEBT_VEBT,
                            newnode2:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),l),
                            $let(
                              newlist: list_VEBT_VEBT,
                              newlist:= list_u1324408373059187874T_VEBT(TreeLista,h,newnode2),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode2))),
                                $ite(
                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode2),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_d_e_l_e_t_e(Summarya),h))),
                                    $let(
                                      sn: vEBT_VEBT,
                                      sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summarya),h),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                        $ite(
                                          xn = Maa,
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                            $let(
                                              maxs: option_nat,
                                              maxs:= vEBT_vebt_maxt(sn),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) )),
                                          one_one_nat )) )),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                    $ite(xn = Maa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h))),one_one_nat)) )) ) )),
                        one_one_nat ) ) ) )) )) )) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_5704_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_d_e_l_e_t_e(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Ya = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Ya = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Ya = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary) )
                   => ( ( Ya = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList2),Summary) )
                     => ( ( Ya = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList2),Summary),Xaa)) ) )
                 => ( ! [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary) )
                       => ( ( Ya = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                    | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                      $ite(
                                        ( ( Xaa = Mi )
                                        & ( Xaa = Ma ) ),
                                        aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,one))))),
                                                  $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Summary)),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary)))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,one)))),one_one_nat))),one_one_nat)),
                                          $let(
                                            xn: nat,
                                            xn:= 
                                              $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                            $let(
                                              l: nat,
                                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                              $let(
                                                h: nat,
                                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                                $ite(
                                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_d_e_l_e_t_e(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l))),
                                                    $let(
                                                      newnode2: vEBT_VEBT,
                                                      newnode2:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l),
                                                      $let(
                                                        newlist: list_VEBT_VEBT,
                                                        newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode2),
                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,newnode2))),
                                                          $ite(
                                                            aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode2),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_d_e_l_e_t_e(Summary),h))),
                                                              $let(
                                                                sn: vEBT_VEBT,
                                                                sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),h),
                                                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                                  $ite(
                                                                    xn = Ma,
                                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,sn))),
                                                                      $let(
                                                                        maxs: option_nat,
                                                                        maxs:= vEBT_vebt_maxt(sn),
                                                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                                                          $ite(maxs = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) )),
                                                                    one_one_nat )) )),
                                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                                                              $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h))),one_one_nat)) )) ) )),
                                                  one_one_nat ) ) ) )) )) )) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T8441311223069195367_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_5705_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xc),Xaa)
     => ( ! [Uu2: $o,Uv: $o] : Xc != aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv))
       => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va: list_VEBT_VEBT,Vb: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList2),Vc)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd)
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_5706_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xc),Xaa)
      <=> (Ya) )
     => ( ( ? [Uu2: $o,Uv: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv))
         => (Ya) )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy)
           => (Ya) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va: list_VEBT_VEBT,Vb: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb)
               => ( (Ya)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList2),Vc)
                 => ( (Ya)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd)
                   => ( (Ya)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_5707_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xc),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A3),
                $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw2)
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3)
               => $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_5708_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xc),Xaa)
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A3),
                  $ite(Xaa = one_one_nat,(B3),$false) ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3)
             => ~ $let(
                    pos: nat,
                    pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_5709_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xc),Xaa)
      <=> (Ya) )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => ( (Ya)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw2)
           => (Ya) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3)
               => ( (Ya)
                <=> ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_5710_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xc: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb)),Xc) ).

% buildup_nothing_in_min_max
tff(fact_5711_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xc: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Nb)),Xc) ).

% buildup_nothing_in_leaf
tff(fact_5712_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V8194947554948674370ptions,Ta),Xc)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Ta),Xc)
        | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Ta),Xc) ) ) ).

% both_member_options_def
tff(fact_5713_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Tree),Nb)
     => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Tree),Xc)
       => ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Tree),Xc)
          | aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Tree),Xc) ) ) ) ).

% member_valid_both_member_options
tff(fact_5714_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o,Uw: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_o_VEBT_VEBT(vEBT_Leaf((Uu)),(Uv2))),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_5715_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,zero_zero_nat,Uv2),Uw)),Ux2) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_5716_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] : ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux2),Uy2)),Uz2) ).

% VEBT_internal.membermima.simps(2)
tff(fact_5717_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))),Xc)
    <=> $ite(
          Xc = zero_zero_nat,
          (A2),
          $ite(Xc = one_one_nat,(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_5718_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: list_VEBT_VEBT,Vb2: vEBT_VEBT,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),zero_zero_nat,Vaa),Vb2)),Xc)
    <=> ( ( Xc = Mia )
        | ( Xc = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_5719_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd2: vEBT_VEBT,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista),Vd2)),Xc)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_5720_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S2: vEBT_VEBT,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy2,aa_nat_nat(suc,V),TreeLista),S2)),Xc)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_5721_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc2: vEBT_VEBT,Xc: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,V),TreeLista),Vc2)),Xc)
    <=> ( ( Xc = Mia )
        | ( Xc = Maa )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_5722_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xc),Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va: list_VEBT_VEBT,Vb: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList2),Vc)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd)
               => ~ $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_5723_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(B3)),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw2: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uv)),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Ya = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uv)),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz) )
                 => ( ( Ya = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd) )
                   => ( ( Ya = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Ya = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                  one_one_nat,
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),
                                        $ite(
                                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),
                                            $let(
                                              maxlow: option_nat,
                                              maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),
                                                $ite(
                                                  ( ( maxlow != none_nat )
                                                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_s_u_c_c(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                                  $let(
                                                    sc: option_nat,
                                                    sc:= vEBT_vebt_succ(Summary,h),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c(Summary),h))),one_one_nat)),
                                                      $ite(sc = none_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(sc)))))) ) )) )),
                                          one_one_nat )) ) ) )) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_5724_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Ya = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite((B3),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va) )
                   => ( ( Ya = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd),Ve) )
                     => ( ( Ya = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd),Ve),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Ya = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                    one_one_nat,
                                    $let(
                                      l: nat,
                                      l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $let(
                                        h: nat,
                                        h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),one_one_nat)),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            $let(
                                              minlow: option_nat,
                                              minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)))),
                                                $ite(
                                                  ( ( minlow != none_nat )
                                                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_T_p_r_e_d(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                                  $let(
                                                    pr: option_nat,
                                                    pr:= vEBT_vebt_pred(Summary,h),
                                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d(Summary),h))),one_one_nat)),
                                                      $ite(pr = none_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(pr)))))) ) )) ),
                                            one_one_nat )) ) ) )) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel2,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_5725_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Ya = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Ya = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Ya = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary) )
                   => ( ( Ya = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList2),Summary) )
                     => ( ( Ya = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TreeList2),Summary),Xaa)) ) )
                 => ( ! [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary) )
                       => ( ( Ya = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),TreeList2),Summary),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Ya = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                                  one_one_nat,
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma ) ),
                                    one_one_nat,
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        l: nat,
                                        l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V1232361888498592333_e_t_e(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                              $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),aa_nat_nat(vEBT_V1232361888498592333_e_t_e(Summary),h),one_one_nat)),
                                            one_one_nat ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_V6368547301243506412_e_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_5726_vebt__delete_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_delete(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Ya = aa_o_VEBT_VEBT(vEBT_Leaf($false),(B3)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Ya = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),$false) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                     => ( ( Ya = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary) )
                   => ( ( Ya = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Deg2,TreeList2),Summary),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2),Smry2) )
                     => ( ( Ya = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2),Smry2) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,TrLst2),Smry2),Xaa)) ) )
                 => ( ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                       => ( ( Ya = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,zero_zero_nat),Tr2),Sm2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Ya = $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi)
                                  | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa) ),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma ) ),
                                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summary))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi,xn,Mi),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                            $let(
                                              newnode2: vEBT_VEBT,
                                              newnode2:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),
                                              $let(
                                                newlist: list_VEBT_VEBT,
                                                newlist:= list_u1324408373059187874T_VEBT(TreeList2,h,newnode2),
                                                $ite(
                                                  aa_VEBT_VEBT_o(vEBT_VEBT_minNull,newnode2),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= aa_nat_VEBT_VEBT(vEBT_vebt_delete(Summary),h),
                                                    aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                            $ite(
                                                              xn = Ma,
                                                              $let(
                                                                maxs: option_nat,
                                                                maxs:= vEBT_vebt_maxt(sn),
                                                                $ite(maxs = none_nat,minn,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),the_nat(maxs))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),the_nat(maxs)))))) ),
                                                              Ma ))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),sn) ),
                                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(minn),
                                                          $ite(xn = Ma,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,h),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(newlist),h)))),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),newlist),Summary) ) ) ),
                                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) ) ) ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_delete_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_5727_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_s_u_c_c2(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Ya = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(B3)),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw2: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uv)),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Ya = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uv)),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz) )
                 => ( ( Ya = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd) )
                   => ( ( Ya = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Ya = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( ( Ya = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                one_one_nat,
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_s_u_c_c2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_s_u_c_c2(Summary),h)),one_one_nat) ) ),
                                      one_one_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_s_u_c_c_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_5728_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
               => ( ( Ya = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                 => ( ( Ya = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) )
                   => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one)))))),
                              $let(
                                xn: nat,
                                xn:= 
                                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                    & ~ ( ( Xaa = Mi )
                                        | ( Xaa = Ma ) ) ),
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)))),
                                      $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t(Summary),h),one_one_nat)),
                                    one_one_nat ) ) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T9217963907923527482_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_5729_vebt__insert_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_insert(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = $ite(
                      Xaa = zero_zero_nat,
                      aa_o_VEBT_VEBT(vEBT_Leaf($true),(B3)),
                      $ite(Xaa = one_one_nat,aa_o_VEBT_VEBT(vEBT_Leaf((A3)),$true),aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
               => ( ( Ya = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                 => ( ( Ya = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) )
                   => ( ( Ya = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( Ya = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),list_u1324408373059187874T_VEBT(TreeList2,h,aa_nat_VEBT_VEBT(vEBT_vebt_insert(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_VEBT_VEBT(vEBT_vebt_insert(Summary),h),Summary)),
                                  aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_5730_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_p_r_e_d2(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Ya = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Ya = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Ya = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va) )
                   => ( ( Ya = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd),Ve) )
                     => ( ( Ya = one_one_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd),Ve),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Ya = one_one_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Ya = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  one_one_nat,
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_p_r_e_d2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),l)),
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_p_r_e_d2(Summary),h)),one_one_nat) ) ),
                                        one_one_nat ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_T_p_r_e_d_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_5731_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3) )
               => ( ( Ya = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,zero_zero_nat,Ts),S3),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3) )
                 => ( ( Ya = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts),S3),Xaa)) ) )
             => ( ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary) )
                   => ( ( Ya = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V2)),TreeList2),Summary),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( Ya = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_T_i_n_s_e_r_t2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(vEBT_T_i_n_s_e_r_t2(Summary),h),one_one_nat)),
                                  one_one_nat ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5076183648494686801_t_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_5732_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $ite(Xaa = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2) )
               => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz) )
                 => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc) )
                   => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              $ite(
                                Xaa = Mi,
                                one_one_nat,
                                aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                  $ite(
                                    Xaa = Ma,
                                    one_one_nat,
                                    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                        one_one_nat,
                                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                                          $ite(
                                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                            one_one_nat,
                                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),
                                              $let(
                                                h: nat,
                                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(vEBT_T_m_e_m_b_e_r(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),one_one_nat) )) )) )) )) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T5837161174952499735_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_5733_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_T_m_e_m_b_e_r2(Xc),Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = one_one_nat )
               => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2) )
               => ( ( Ya = one_one_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz) )
                 => ( ( Ya = one_one_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc) )
                   => ( ( Ya = one_one_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                              $ite(
                                Xaa = Mi,
                                zero_zero_nat,
                                $ite(
                                  Xaa = Ma,
                                  zero_zero_nat,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                    zero_zero_nat,
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                      zero_zero_nat,
                                      $ite(
                                        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa)
                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Ma) ),
                                        $let(
                                          h: nat,
                                          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_nat(vEBT_T_m_e_m_b_e_r2(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_nat) ),
                                        zero_zero_nat ) ) ) ) )) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_T8099345112685741742_r_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_5734_vebt__member_Opelims_I3_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xc),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2),Xaa)) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz),Xaa)) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_5735_vebt__member_Opelims_I2_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xc),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_5736_vebt__member_Opelims_I1_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Xc),Xaa)
      <=> (Ya) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( (Ya)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),Xaa)) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2) )
               => ( ~ (Ya)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2),Xaa)) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz) )
                 => ( ~ (Ya)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Uy),Uz),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc) )
                   => ( ~ (Ya)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vb),Vc),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( ( (Ya)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_5737_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xc),Xaa)
      <=> (Ya) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( (Ya)
                <=> $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),Xaa)) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw2) )
               => ( ~ (Ya)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw2),Xaa)) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3) )
                 => ( ( (Ya)
                    <=> $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_5738_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xc),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),Xaa))
               => ~ $ite(
                      Xaa = zero_zero_nat,
                      (A3),
                      $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_5739_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,Xc),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),Xaa))
               => $ite(
                    Xaa = zero_zero_nat,
                    (A3),
                    $ite(Xaa = one_one_nat,(B3),$false) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,zero_zero_nat,Uv),Uw2),Xaa)) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uy,aa_nat_nat(suc,V2),TreeList2),S3),Xaa))
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_V5719532721284313246member,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_5740_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xc),Xaa)
      <=> (Ya) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)) )
             => ( ~ (Ya)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)),Xaa)) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy) )
               => ( ~ (Ya)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb) )
                 => ( ( (Ya)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList2),Vc) )
                   => ( ( (Ya)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList2),Vc),Xaa)) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd) )
                     => ( ( (Ya)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_5741_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xc),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)),Xaa)) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux),Uy),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList2),Vc) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList2),Vc),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd),Xaa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_5742_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,Xc),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT] :
              ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va),Vb),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList2),Vc) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V2),TreeList2),Vc),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V2),TreeList2),Vd),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList2)),aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_membermima,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_5743_pred__less__length__list,axiom,
    ! [Deg: nat,Xc: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Maa)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya),Xc) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xc),some_nat(Mia),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_5744_pred__lesseq__max,axiom,
    ! [Deg: nat,Xc: nat,Maa: nat,Mia: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),Maa)
       => ( vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya),Xc) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    minlow: option_nat,
                    minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),
                    $ite(
                      ( ( minlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h),l)),
                      $let(
                        pr: option_nat,
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none_nat,
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xc),some_nat(Mia),none_nat),
                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(pr)))) ) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_5745_succ__greatereq__min,axiom,
    ! [Deg: nat,Mia: nat,Xc: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xc)
       => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya),Xc) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(sc))))) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_5746_succ__less__length__list,axiom,
    ! [Deg: nat,Mia: nat,Xc: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xc)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Deg,TreeLista),Summarya),Xc) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h),l)),
                      $let(
                        sc: option_nat,
                        sc:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_5747_foldr__zero,axiom,
    ! [Xs: list_nat,D2: nat] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_nat(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(nth_nat(Xs),I5)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),D2)),D2)) ) ).

% foldr_zero
tff(fact_5748_add__shift,axiom,
    ! [Xc: nat,Ya: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Ya) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(Xc)),some_nat(Ya)) = some_nat(Z) ) ) ).

% add_shift
tff(fact_5749_mul__shift,axiom,
    ! [Xc: nat,Ya: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xc),Ya) = Z )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(Xc)),some_nat(Ya)) = some_nat(Z) ) ) ).

% mul_shift
tff(fact_5750_foldr__one,axiom,
    ! [D2: nat,Ys: list_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D2)) ).

% foldr_one
tff(fact_5751_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_5752_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_5753_foldr__same__int,axiom,
    ! [Xs: list_nat,Ya: nat] :
      ( ! [X3: nat,Y: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => ( aa_set_nat_o(member_nat(Y),set_nat2(Xs))
           => ( X3 = Y ) ) )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
           => ( X3 = Ya ) )
       => ( aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Ya) ) ) ) ).

% foldr_same_int
tff(fact_5754_foldr__mono,axiom,
    ! [Xs: list_nat,Ys: list_nat,C2: nat,D2: nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( ! [I5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),size_size_list_nat(Xs))
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(nth_nat(Xs),I5)),aa_nat_nat(nth_nat(Ys),I5)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),D2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),C2)),size_size_list_nat(Ys))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,Ys),D2)) ) ) ) ).

% foldr_mono
tff(fact_5755_foldr__length,axiom,
    ! [L: list_VEBT_VEBT] : aa_nat_nat(foldr_VEBT_VEBT_nat(aTP_Lamm_dt,L),zero_zero_nat) = size_s6755466524823107622T_VEBT(L) ).

% foldr_length
tff(fact_5756_foldr__length,axiom,
    ! [L: list_real] : aa_nat_nat(foldr_real_nat(aTP_Lamm_du,L),zero_zero_nat) = size_size_list_real(L) ).

% foldr_length
tff(fact_5757_foldr__length,axiom,
    ! [L: list_o] : aa_nat_nat(foldr_o_nat(aTP_Lamm_dv,L),zero_zero_nat) = size_size_list_o(L) ).

% foldr_length
tff(fact_5758_foldr__length,axiom,
    ! [L: list_nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_dw,L),zero_zero_nat) = size_size_list_nat(L) ).

% foldr_length
tff(fact_5759_foldr__length,axiom,
    ! [L: list_VEBT_VEBTi] : aa_nat_nat(foldr_VEBT_VEBTi_nat(aTP_Lamm_dx,L),zero_zero_nat) = size_s7982070591426661849_VEBTi(L) ).

% foldr_length
tff(fact_5760_foldr__cong,axiom,
    ! [A2: nat,B2: nat,L: list_nat,K: list_nat,F2: fun_nat_fun_nat_nat,G: fun_nat_fun_nat_nat] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A3: nat,X3: nat] :
              ( aa_set_nat_o(member_nat(X3),set_nat2(L))
             => ( aa_nat_nat(aa_nat_fun_nat_nat(F2,X3),A3) = aa_nat_nat(aa_nat_fun_nat_nat(G,X3),A3) ) )
         => ( aa_nat_nat(foldr_nat_nat(F2,L),A2) = aa_nat_nat(foldr_nat_nat(G,K),B2) ) ) ) ) ).

% foldr_cong
tff(fact_5761_foldr__cong,axiom,
    ! [A2: real,B2: real,L: list_real,K: list_real,F2: fun_re7681423839302404990l_real,G: fun_re7681423839302404990l_real] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A3: real,X3: real] :
              ( aa_set_real_o(member_real(X3),set_real2(L))
             => ( aa_real_real(aa_rea7340605660300375871l_real(F2,X3),A3) = aa_real_real(aa_rea7340605660300375871l_real(G,X3),A3) ) )
         => ( aa_real_real(foldr_real_real(F2,L),A2) = aa_real_real(foldr_real_real(G,K),B2) ) ) ) ) ).

% foldr_cong
tff(fact_5762_foldr__length__aux,axiom,
    ! [L: list_VEBT_VEBT,A2: nat] : aa_nat_nat(foldr_VEBT_VEBT_nat(aTP_Lamm_dt,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_s6755466524823107622T_VEBT(L)) ).

% foldr_length_aux
tff(fact_5763_foldr__length__aux,axiom,
    ! [L: list_real,A2: nat] : aa_nat_nat(foldr_real_nat(aTP_Lamm_du,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_size_list_real(L)) ).

% foldr_length_aux
tff(fact_5764_foldr__length__aux,axiom,
    ! [L: list_o,A2: nat] : aa_nat_nat(foldr_o_nat(aTP_Lamm_dv,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_size_list_o(L)) ).

% foldr_length_aux
tff(fact_5765_foldr__length__aux,axiom,
    ! [L: list_nat,A2: nat] : aa_nat_nat(foldr_nat_nat(aTP_Lamm_dw,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_size_list_nat(L)) ).

% foldr_length_aux
tff(fact_5766_foldr__length__aux,axiom,
    ! [L: list_VEBT_VEBTi,A2: nat] : aa_nat_nat(foldr_VEBT_VEBTi_nat(aTP_Lamm_dx,L),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),size_s7982070591426661849_VEBTi(L)) ).

% foldr_length_aux
tff(fact_5767_vebt__succ_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      vEBT_vebt_succ(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya),Xc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Mia),
        some_nat(Mia),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                maxlow: option_nat,
                maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),
                $ite(
                  ( ( maxlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h),l)),
                  $let(
                    sc: option_nat,
                    sc:= vEBT_vebt_succ(Summarya,h),
                    $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(sc))))) ) ) ),
              none_nat ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_5768_vebt__pred_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xc: nat] :
      vEBT_vebt_pred(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),TreeLista),Summarya),Xc) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xc),
        some_nat(Maa),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $let(
            h: nat,
            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
              $let(
                minlow: option_nat,
                minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h)),
                $ite(
                  ( ( minlow != none_nat )
                  & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                  aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),h),l)),
                  $let(
                    pr: option_nat,
                    pr:= vEBT_vebt_pred(Summarya,h),
                    $ite(
                      pr = none_nat,
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xc),some_nat(Mia),none_nat),
                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeLista),the_nat(pr)))) ) ) ) ),
              none_nat ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_5769_vebt__succ_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: option_nat] :
      ( ( vEBT_vebt_succ(Xc,Xaa) = Ya )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(B3)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Ya != $ite((B3),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv: $o,Uw2: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uv)),(Uw2))
           => ( ? [N: nat] : Xaa = aa_nat_nat(suc,N)
             => ( Ya != none_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz)
             => ( Ya != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd)
               => ( Ya != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2)
                 => ( Ya != none_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                     => ( Ya != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            some_nat(Mi),
                            $let(
                              l: nat,
                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h),l)),
                                      $let(
                                        sc: option_nat,
                                        sc:= vEBT_vebt_succ(Summary,h),
                                        $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(sc))))) ) ) ),
                                  none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_5770_vebt__pred_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: option_nat] :
      ( ( vEBT_vebt_pred(Xc,Xaa) = Ya )
     => ( ( ? [Uu2: $o,Uv: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv))
         => ( ( Xaa = zero_zero_nat )
           => ( Ya != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw2: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(Uw2))
             => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
               => ( Ya != $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
               => ( ? [Va2: nat] : Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2))
                 => ( Ya != $ite(
                        (B3),
                        some_nat(one_one_nat),
                        $ite((A3),some_nat(zero_zero_nat),none_nat) ) ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va)
               => ( Ya != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd),Ve)
                 => ( Ya != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2)
                   => ( Ya != none_nat ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( Ya != $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              some_nat(Ma),
                              $let(
                                l: nat,
                                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $let(
                                  h: nat,
                                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                    $let(
                                      minlow: option_nat,
                                      minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none_nat )
                                        & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                        aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h),l)),
                                        $let(
                                          pr: option_nat,
                                          pr:= vEBT_vebt_pred(Summary,h),
                                          $ite(
                                            pr = none_nat,
                                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),some_nat(Mi),none_nat),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(pr)))) ) ) ) ),
                                    none_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_5771_vebt__succ_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: option_nat] :
      ( ( vEBT_vebt_succ(Xc,Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [Uu2: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(B3)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Ya = $ite((B3),some_nat(one_one_nat),none_nat) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(B3)),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw2: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uv)),(Uw2)) )
               => ! [N: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N) )
                   => ( ( Ya = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uv)),(Uw2)),aa_nat_nat(suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz) )
                 => ( ( Ya = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy),Uz),Xaa)) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd) )
                   => ( ( Ya = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vc),Vd),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2) )
                     => ( ( Ya = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vg2),Vh2),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                       => ( ( Ya = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                some_nat(Mi),
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(
                                      aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_less,some_nat(l)),maxlow) ),
                                          aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_succ(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h),l)),
                                          $let(
                                            sc: option_nat,
                                            sc:= vEBT_vebt_succ(Summary,h),
                                            $ite(sc = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),sc)),vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(sc))))) ) ) ),
                                      none_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_5772_vebt__pred_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: nat,Ya: option_nat] :
      ( ( vEBT_vebt_pred(Xc,Xaa) = Ya )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(Xc,Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)) )
             => ( ( Xaa = zero_zero_nat )
               => ( ( Ya = none_nat )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),(Uv)),zero_zero_nat)) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(Uw2)) )
               => ( ( Xaa = aa_nat_nat(suc,zero_zero_nat) )
                 => ( ( Ya = $ite((A3),some_nat(zero_zero_nat),none_nat) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(Uw2)),aa_nat_nat(suc,zero_zero_nat))) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                     => ( ( Ya = $ite(
                              (B3),
                              some_nat(one_one_nat),
                              $ite((A3),some_nat(zero_zero_nat),none_nat) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)))) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va) )
                   => ( ( Ya = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uy,Uz),Va),Xaa)) ) )
               => ( ! [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd),Ve) )
                     => ( ( Ya = none_nat )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),zero_zero_nat,Vd),Ve),Xaa)) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2) )
                       => ( ( Ya = none_nat )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(V2),aa_nat_nat(suc,zero_zero_nat),Vh2),Vi2),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary) )
                         => ( ( Ya = $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  some_nat(Ma),
                                  $let(
                                    l: nat,
                                    l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(
                                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList2)),
                                        $let(
                                          minlow: option_nat,
                                          minlow:= vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none_nat )
                                            & aa_option_nat_o(aa_opt4319133605541840243_nat_o(vEBT_VEBT_greater,some_nat(l)),minlow) ),
                                            aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(h))),vEBT_vebt_pred(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),h),l)),
                                            $let(
                                              pr: option_nat,
                                              pr:= vEBT_vebt_pred(Summary,h),
                                              $ite(
                                                pr = none_nat,
                                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),Xaa),some_nat(Mi),none_nat),
                                                aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),pr)),vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(TreeList2),the_nat(pr)))) ) ) ) ),
                                        none_nat ) ) ) ) )
                           => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_pred_rel,produc738532404422230701BT_nat(aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeList2),Summary),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_5773_lowi__hT,axiom,
    ! [Xc: nat,Nb: nat] : aa_nat_o(time_htt_nat(one_one_assn,vEBT_VEBT_lowi(Xc,Nb),aa_nat_fun_nat_assn(aTP_Lamm_dy(Xc),Nb)),one_one_nat) ).

% lowi_hT
tff(fact_5774_highi__hT,axiom,
    ! [Xc: nat,Nb: nat] : aa_nat_o(time_htt_nat(one_one_assn,vEBT_VEBT_highi(Xc,Nb),aa_nat_fun_nat_assn(aTP_Lamm_dz(Xc),Nb)),one_one_nat) ).

% highi_hT
tff(fact_5775_foldr__same,axiom,
    ! [Xs: list_real,Ya: real] :
      ( ! [X3: real,Y: real] :
          ( aa_set_real_o(member_real(X3),set_real2(Xs))
         => ( aa_set_real_o(member_real(Y),set_real2(Xs))
           => ( X3 = Y ) ) )
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),set_real2(Xs))
           => ( X3 = Ya ) )
       => ( aa_real_real(foldr_real_real(plus_plus_real,Xs),zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Ya) ) ) ) ).

% foldr_same
tff(fact_5776_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F2: fun_VEBT_VEBT_nat,Bound: nat,I: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(F2,X3)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F2,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5777_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_real,F2: fun_real_nat,Bound: nat,I: nat] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_real2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_real_nat(F2,X3)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_real_nat(F2,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5778_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_o,F2: fun_o_nat,Bound: nat,I: nat] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),set_o2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_o_nat(F2,(X3))),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_o_nat(F2,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5779_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_nat,F2: fun_nat_nat,Bound: nat,I: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F2,X3)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_nat_nat(F2,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5780_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs: list_VEBT_VEBTi,F2: fun_VEBT_VEBTi_nat,Bound: nat,I: nat] :
      ( ! [X3: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(Xs))
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBTi_nat(F2,X3)),Bound) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBTi_nat(F2,Xs)),I)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s7982070591426661849_VEBTi(Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_5781_TBOUND__highi,axiom,
    ! [Xc: nat,Nb: nat] : aa_nat_o(time_TBOUND_nat(vEBT_VEBT_highi(Xc,Nb)),one_one_nat) ).

% TBOUND_highi
tff(fact_5782_TBOUND__lowi,axiom,
    ! [Xc: nat,Nb: nat] : aa_nat_o(time_TBOUND_nat(vEBT_VEBT_lowi(Xc,Nb)),one_one_nat) ).

% TBOUND_lowi
tff(fact_5783_foldr0,axiom,
    ! [Xs: list_real,C2: real,D2: real] : aa_real_real(foldr_real_real(plus_plus_real,Xs),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),D2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(foldr_real_real(plus_plus_real,Xs),D2)),C2) ).

% foldr0
tff(fact_5784_highi__h,axiom,
    ! [Xc: nat,Nb: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_highi(Xc,Nb),aa_nat_fun_nat_assn(aTP_Lamm_dz(Xc),Nb)) ).

% highi_h
tff(fact_5785_lowi__h,axiom,
    ! [Xc: nat,Nb: nat] : hoare_3067605981109127869le_nat(one_one_assn,vEBT_VEBT_lowi(Xc,Nb),aa_nat_fun_nat_assn(aTP_Lamm_dy(Xc),Nb)) ).

% lowi_h
tff(fact_5786_length__map,axiom,
    ! [F2: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT] : size_s6755466524823107622T_VEBT(map_VE8901447254227204932T_VEBT(F2,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_5787_length__map,axiom,
    ! [F2: fun_real_VEBT_VEBT,Xs: list_real] : size_s6755466524823107622T_VEBT(map_real_VEBT_VEBT(F2,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_5788_length__map,axiom,
    ! [F2: fun_o_VEBT_VEBT,Xs: list_o] : size_s6755466524823107622T_VEBT(map_o_VEBT_VEBT(F2,Xs)) = size_size_list_o(Xs) ).

% length_map
tff(fact_5789_length__map,axiom,
    ! [F2: fun_nat_VEBT_VEBT,Xs: list_nat] : size_s6755466524823107622T_VEBT(map_nat_VEBT_VEBT(F2,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_5790_length__map,axiom,
    ! [F2: fun_VE9120655849212850072T_VEBT,Xs: list_VEBT_VEBTi] : size_s6755466524823107622T_VEBT(map_VE7998069337340375161T_VEBT(F2,Xs)) = size_s7982070591426661849_VEBTi(Xs) ).

% length_map
tff(fact_5791_length__map,axiom,
    ! [F2: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT] : size_size_list_real(map_VEBT_VEBT_real(F2,Xs)) = size_s6755466524823107622T_VEBT(Xs) ).

% length_map
tff(fact_5792_length__map,axiom,
    ! [F2: fun_real_real,Xs: list_real] : size_size_list_real(map_real_real(F2,Xs)) = size_size_list_real(Xs) ).

% length_map
tff(fact_5793_length__map,axiom,
    ! [F2: fun_o_real,Xs: list_o] : size_size_list_real(map_o_real(F2,Xs)) = size_size_list_o(Xs) ).

% length_map
tff(fact_5794_length__map,axiom,
    ! [F2: fun_nat_real,Xs: list_nat] : size_size_list_real(map_nat_real(F2,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_5795_length__map,axiom,
    ! [F2: fun_VEBT_VEBTi_real,Xs: list_VEBT_VEBTi] : size_size_list_real(map_VEBT_VEBTi_real(F2,Xs)) = size_s7982070591426661849_VEBTi(Xs) ).

% length_map
tff(fact_5796_map__eq__conv,axiom,
    ! [F2: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat] :
      ( ( map_VEBT_VEBT_nat(F2,Xs) = map_VEBT_VEBT_nat(G,Xs) )
    <=> ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_nat(F2,X2) = aa_VEBT_VEBT_nat(G,X2) ) ) ) ).

% map_eq_conv
tff(fact_5797_map__eq__conv,axiom,
    ! [F2: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real] :
      ( ( map_VEBT_VEBT_real(F2,Xs) = map_VEBT_VEBT_real(G,Xs) )
    <=> ! [X2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X2),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_real(F2,X2) = aa_VEBT_VEBT_real(G,X2) ) ) ) ).

% map_eq_conv
tff(fact_5798_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F2: fun_VE3165209603783583043T_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(map_VE8901447254227204932T_VEBT(F2,Xs)),Nb) = aa_VEB5832953974749077948T_VEBT(F2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)) ) ) ).

% nth_map
tff(fact_5799_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F2: fun_VE712148701811721852_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( nth_VEBT_VEBTi(map_VE7029150624388687525_VEBTi(F2,Xs),Nb) = aa_VEB3973023575446460973_VEBTi(F2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)) ) ) ).

% nth_map
tff(fact_5800_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F2: fun_VEBT_VEBT_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_nat(nth_nat(map_VEBT_VEBT_nat(F2,Xs)),Nb) = aa_VEBT_VEBT_nat(F2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)) ) ) ).

% nth_map
tff(fact_5801_nth__map,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,F2: fun_VEBT_VEBT_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s6755466524823107622T_VEBT(Xs))
     => ( aa_nat_real(nth_real(map_VEBT_VEBT_real(F2,Xs)),Nb) = aa_VEBT_VEBT_real(F2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),Nb)) ) ) ).

% nth_map
tff(fact_5802_nth__map,axiom,
    ! [Nb: nat,Xs: list_real,F2: fun_real_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(map_real_VEBT_VEBT(F2,Xs)),Nb) = aa_real_VEBT_VEBT(F2,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% nth_map
tff(fact_5803_nth__map,axiom,
    ! [Nb: nat,Xs: list_real,F2: fun_real_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => ( nth_VEBT_VEBTi(map_real_VEBT_VEBTi(F2,Xs),Nb) = aa_real_VEBT_VEBTi(F2,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% nth_map
tff(fact_5804_nth__map,axiom,
    ! [Nb: nat,Xs: list_real,F2: fun_real_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => ( aa_nat_nat(nth_nat(map_real_nat(F2,Xs)),Nb) = aa_real_nat(F2,aa_nat_real(nth_real(Xs),Nb)) ) ) ).

% nth_map
tff(fact_5805_nth__map,axiom,
    ! [Nb: nat,Xs: list_o,F2: fun_o_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( aa_nat_VEBT_VEBT(nth_VEBT_VEBT(map_o_VEBT_VEBT(F2,Xs)),Nb) = aa_o_VEBT_VEBT(F2,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% nth_map
tff(fact_5806_nth__map,axiom,
    ! [Nb: nat,Xs: list_o,F2: fun_o_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( nth_VEBT_VEBTi(map_o_VEBT_VEBTi(F2,Xs),Nb) = aa_o_VEBT_VEBTi(F2,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% nth_map
tff(fact_5807_nth__map,axiom,
    ! [Nb: nat,Xs: list_o,F2: fun_o_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => ( aa_nat_nat(nth_nat(map_o_nat(F2,Xs)),Nb) = aa_o_nat(F2,aa_nat_o(nth_o(Xs),Nb)) ) ) ).

% nth_map
tff(fact_5808_map__eq__imp__length__eq,axiom,
    ! [F2: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_nat,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat(F2,Xs) = map_VEBT_VEBT_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5809_map__eq__imp__length__eq,axiom,
    ! [F2: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBT_real,Ys: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_real(F2,Xs) = map_VEBT_VEBT_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5810_map__eq__imp__length__eq,axiom,
    ! [F2: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_real_nat,Ys: list_real] :
      ( ( map_VEBT_VEBT_nat(F2,Xs) = map_real_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5811_map__eq__imp__length__eq,axiom,
    ! [F2: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_real_real,Ys: list_real] :
      ( ( map_VEBT_VEBT_real(F2,Xs) = map_real_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_real(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5812_map__eq__imp__length__eq,axiom,
    ! [F2: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_o_nat,Ys: list_o] :
      ( ( map_VEBT_VEBT_nat(F2,Xs) = map_o_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5813_map__eq__imp__length__eq,axiom,
    ! [F2: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_o_real,Ys: list_o] :
      ( ( map_VEBT_VEBT_real(F2,Xs) = map_o_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_o(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5814_map__eq__imp__length__eq,axiom,
    ! [F2: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_nat_nat,Ys: list_nat] :
      ( ( map_VEBT_VEBT_nat(F2,Xs) = map_nat_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5815_map__eq__imp__length__eq,axiom,
    ! [F2: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_nat_real,Ys: list_nat] :
      ( ( map_VEBT_VEBT_real(F2,Xs) = map_nat_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5816_map__eq__imp__length__eq,axiom,
    ! [F2: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBTi_nat,Ys: list_VEBT_VEBTi] :
      ( ( map_VEBT_VEBT_nat(F2,Xs) = map_VEBT_VEBTi_nat(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s7982070591426661849_VEBTi(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5817_map__eq__imp__length__eq,axiom,
    ! [F2: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,G: fun_VEBT_VEBTi_real,Ys: list_VEBT_VEBTi] :
      ( ( map_VEBT_VEBT_real(F2,Xs) = map_VEBT_VEBTi_real(G,Ys) )
     => ( size_s6755466524823107622T_VEBT(Xs) = size_s7982070591426661849_VEBTi(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5818_list_Omap__cong,axiom,
    ! [Xc: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F2: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ( Xc = Ya )
     => ( ! [Z2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Ya))
           => ( aa_VEBT_VEBT_nat(F2,Z2) = aa_VEBT_VEBT_nat(G,Z2) ) )
       => ( map_VEBT_VEBT_nat(F2,Xc) = map_VEBT_VEBT_nat(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5819_list_Omap__cong,axiom,
    ! [Xc: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F2: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ( Xc = Ya )
     => ( ! [Z2: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Ya))
           => ( aa_VEBT_VEBT_real(F2,Z2) = aa_VEBT_VEBT_real(G,Z2) ) )
       => ( map_VEBT_VEBT_real(F2,Xc) = map_VEBT_VEBT_real(G,Ya) ) ) ) ).

% list.map_cong
tff(fact_5820_list_Omap__cong0,axiom,
    ! [Xc: list_VEBT_VEBT,F2: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [Z2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Xc))
         => ( aa_VEBT_VEBT_nat(F2,Z2) = aa_VEBT_VEBT_nat(G,Z2) ) )
     => ( map_VEBT_VEBT_nat(F2,Xc) = map_VEBT_VEBT_nat(G,Xc) ) ) ).

% list.map_cong0
tff(fact_5821_list_Omap__cong0,axiom,
    ! [Xc: list_VEBT_VEBT,F2: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [Z2: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Xc))
         => ( aa_VEBT_VEBT_real(F2,Z2) = aa_VEBT_VEBT_real(G,Z2) ) )
     => ( map_VEBT_VEBT_real(F2,Xc) = map_VEBT_VEBT_real(G,Xc) ) ) ).

% list.map_cong0
tff(fact_5822_list_Oinj__map__strong,axiom,
    ! [Xc: list_VEBT_VEBT,Xaa: list_VEBT_VEBT,F2: fun_VEBT_VEBT_nat,Fa: fun_VEBT_VEBT_nat] :
      ( ! [Z2: vEBT_VEBT,Za: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Xc))
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Za),set_VEBT_VEBT2(Xaa))
           => ( ( aa_VEBT_VEBT_nat(F2,Z2) = aa_VEBT_VEBT_nat(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( map_VEBT_VEBT_nat(F2,Xc) = map_VEBT_VEBT_nat(Fa,Xaa) )
       => ( Xc = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_5823_list_Oinj__map__strong,axiom,
    ! [Xc: list_VEBT_VEBT,Xaa: list_VEBT_VEBT,F2: fun_VEBT_VEBT_real,Fa: fun_VEBT_VEBT_real] :
      ( ! [Z2: vEBT_VEBT,Za: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Z2),set_VEBT_VEBT2(Xc))
         => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Za),set_VEBT_VEBT2(Xaa))
           => ( ( aa_VEBT_VEBT_real(F2,Z2) = aa_VEBT_VEBT_real(Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( map_VEBT_VEBT_real(F2,Xc) = map_VEBT_VEBT_real(Fa,Xaa) )
       => ( Xc = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_5824_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F2: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_nat(F2,X3) = aa_VEBT_VEBT_nat(G,X3) ) )
     => ( map_VEBT_VEBT_nat(F2,Xs) = map_VEBT_VEBT_nat(G,Xs) ) ) ).

% map_ext
tff(fact_5825_map__ext,axiom,
    ! [Xs: list_VEBT_VEBT,F2: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs))
         => ( aa_VEBT_VEBT_real(F2,X3) = aa_VEBT_VEBT_real(G,X3) ) )
     => ( map_VEBT_VEBT_real(F2,Xs) = map_VEBT_VEBT_real(G,Xs) ) ) ).

% map_ext
tff(fact_5826_map__idI,axiom,
    ! [Xs: list_int,F2: fun_int_int] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),set_int2(Xs))
         => ( aa_int_int(F2,X3) = X3 ) )
     => ( map_int_int(F2,Xs) = Xs ) ) ).

% map_idI
tff(fact_5827_map__idI,axiom,
    ! [Xs: list_complex,F2: fun_complex_complex] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),set_complex2(Xs))
         => ( aa_complex_complex(F2,X3) = X3 ) )
     => ( map_complex_complex(F2,Xs) = Xs ) ) ).

% map_idI
tff(fact_5828_map__idI,axiom,
    ! [Xs: list_VEBT_VEBT,F2: fun_VE3165209603783583043T_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs))
         => ( aa_VEB5832953974749077948T_VEBT(F2,X3) = X3 ) )
     => ( map_VE8901447254227204932T_VEBT(F2,Xs) = Xs ) ) ).

% map_idI
tff(fact_5829_map__idI,axiom,
    ! [Xs: list_nat,F2: fun_nat_nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => ( aa_nat_nat(F2,X3) = X3 ) )
     => ( map_nat_nat(F2,Xs) = Xs ) ) ).

% map_idI
tff(fact_5830_map__idI,axiom,
    ! [Xs: list_real,F2: fun_real_real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_real2(Xs))
         => ( aa_real_real(F2,X3) = X3 ) )
     => ( map_real_real(F2,Xs) = Xs ) ) ).

% map_idI
tff(fact_5831_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F2: fun_VEBT_VEBT_nat,G: fun_VEBT_VEBT_nat] :
      ( ( Xs = Ys )
     => ( ! [X3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Ys))
           => ( aa_VEBT_VEBT_nat(F2,X3) = aa_VEBT_VEBT_nat(G,X3) ) )
       => ( map_VEBT_VEBT_nat(F2,Xs) = map_VEBT_VEBT_nat(G,Ys) ) ) ) ).

% map_cong
tff(fact_5832_map__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F2: fun_VEBT_VEBT_real,G: fun_VEBT_VEBT_real] :
      ( ( Xs = Ys )
     => ( ! [X3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Ys))
           => ( aa_VEBT_VEBT_real(F2,X3) = aa_VEBT_VEBT_real(G,X3) ) )
       => ( map_VEBT_VEBT_real(F2,Xs) = map_VEBT_VEBT_real(G,Ys) ) ) ) ).

% map_cong
tff(fact_5833_ex__map__conv,axiom,
    ! [Ys: list_nat,F2: fun_VEBT_VEBT_nat] :
      ( ? [Xs3: list_VEBT_VEBT] : Ys = map_VEBT_VEBT_nat(F2,Xs3)
    <=> ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),set_nat2(Ys))
         => ? [Xa3: vEBT_VEBT] : X2 = aa_VEBT_VEBT_nat(F2,Xa3) ) ) ).

% ex_map_conv
tff(fact_5834_ex__map__conv,axiom,
    ! [Ys: list_real,F2: fun_VEBT_VEBT_real] :
      ( ? [Xs3: list_VEBT_VEBT] : Ys = map_VEBT_VEBT_real(F2,Xs3)
    <=> ! [X2: real] :
          ( aa_set_real_o(member_real(X2),set_real2(Ys))
         => ? [Xa3: vEBT_VEBT] : X2 = aa_VEBT_VEBT_real(F2,Xa3) ) ) ).

% ex_map_conv
tff(fact_5835_map__eq__nth__eq,axiom,
    ! [F2: fun_VEBT_VEBT_nat,L: list_VEBT_VEBT,L2: list_VEBT_VEBT,I: nat] :
      ( ( map_VEBT_VEBT_nat(F2,L) = map_VEBT_VEBT_nat(F2,L2) )
     => ( aa_VEBT_VEBT_nat(F2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_nat(F2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L2),I)) ) ) ).

% map_eq_nth_eq
tff(fact_5836_map__eq__nth__eq,axiom,
    ! [F2: fun_VEBT_VEBT_real,L: list_VEBT_VEBT,L2: list_VEBT_VEBT,I: nat] :
      ( ( map_VEBT_VEBT_real(F2,L) = map_VEBT_VEBT_real(F2,L2) )
     => ( aa_VEBT_VEBT_real(F2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_real(F2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L2),I)) ) ) ).

% map_eq_nth_eq
tff(fact_5837_map__update,axiom,
    ! [F2: fun_VE8449866691444886503_VEBTi,Xs: list_VEBT_VEBTi,K: nat,Ya: vEBT_VEBTi] : map_VE483055756984248624_VEBTi(F2,list_u6098035379799741383_VEBTi(Xs,K,Ya)) = list_u6098035379799741383_VEBTi(map_VE483055756984248624_VEBTi(F2,Xs),K,aa_VEB5586453056816421800_VEBTi(F2,Ya)) ).

% map_update
tff(fact_5838_map__update,axiom,
    ! [F2: fun_VE9120655849212850072T_VEBT,Xs: list_VEBT_VEBTi,K: nat,Ya: vEBT_VEBTi] : map_VE7998069337340375161T_VEBT(F2,list_u6098035379799741383_VEBTi(Xs,K,Ya)) = list_u1324408373059187874T_VEBT(map_VE7998069337340375161T_VEBT(F2,Xs),K,aa_VEB4941942288398148609T_VEBT(F2,Ya)) ).

% map_update
tff(fact_5839_map__update,axiom,
    ! [F2: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,K: nat,Ya: vEBT_VEBT] : map_VEBT_VEBT_nat(F2,list_u1324408373059187874T_VEBT(Xs,K,Ya)) = list_update_nat(map_VEBT_VEBT_nat(F2,Xs),K,aa_VEBT_VEBT_nat(F2,Ya)) ).

% map_update
tff(fact_5840_map__update,axiom,
    ! [F2: fun_VEBT_VEBT_real,Xs: list_VEBT_VEBT,K: nat,Ya: vEBT_VEBT] : map_VEBT_VEBT_real(F2,list_u1324408373059187874T_VEBT(Xs,K,Ya)) = list_update_real(map_VEBT_VEBT_real(F2,Xs),K,aa_VEBT_VEBT_real(F2,Ya)) ).

% map_update
tff(fact_5841_map__update,axiom,
    ! [F2: fun_VE712148701811721852_VEBTi,Xs: list_VEBT_VEBT,K: nat,Ya: vEBT_VEBT] : map_VE7029150624388687525_VEBTi(F2,list_u1324408373059187874T_VEBT(Xs,K,Ya)) = list_u6098035379799741383_VEBTi(map_VE7029150624388687525_VEBTi(F2,Xs),K,aa_VEB3973023575446460973_VEBTi(F2,Ya)) ).

% map_update
tff(fact_5842_map__update,axiom,
    ! [F2: fun_VE3165209603783583043T_VEBT,Xs: list_VEBT_VEBT,K: nat,Ya: vEBT_VEBT] : map_VE8901447254227204932T_VEBT(F2,list_u1324408373059187874T_VEBT(Xs,K,Ya)) = list_u1324408373059187874T_VEBT(map_VE8901447254227204932T_VEBT(F2,Xs),K,aa_VEB5832953974749077948T_VEBT(F2,Ya)) ).

% map_update
tff(fact_5843_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F2: fun_VEBT_VEBT_nat,Xc: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
       => ( aa_VEBT_VEBT_nat(F2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_nat(F2,Xc) ) )
     => ( map_VEBT_VEBT_nat(F2,list_u1324408373059187874T_VEBT(L,I,Xc)) = map_VEBT_VEBT_nat(F2,L) ) ) ).

% map_upd_eq
tff(fact_5844_map__upd__eq,axiom,
    ! [I: nat,L: list_VEBT_VEBT,F2: fun_VEBT_VEBT_real,Xc: vEBT_VEBT] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(L))
       => ( aa_VEBT_VEBT_real(F2,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(L),I)) = aa_VEBT_VEBT_real(F2,Xc) ) )
     => ( map_VEBT_VEBT_real(F2,list_u1324408373059187874T_VEBT(L,I,Xc)) = map_VEBT_VEBT_real(F2,L) ) ) ).

% map_upd_eq
tff(fact_5845_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summarya))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeLista)),zero_zero_nat)) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_5846_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xc) = Ya )
     => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( Ya != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary) )
             => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_5847_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space2,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summarya))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeLista)),zero_zero_nat)) ).

% VEBT_internal.space'.simps(2)
tff(fact_5848_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xc) = Ya )
     => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary) )
             => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_5849_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_space,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summarya))),size_s6755466524823107622T_VEBT(TreeLista))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeLista)),zero_zero_nat)) ).

% VEBT_internal.space.simps(2)
tff(fact_5850_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xc) = Ya )
     => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary) )
             => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_5851_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_VEBT_VEBT,F2: fun_VEBT_VEBT_real,Bound: real,I: real] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F2,X3)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F2,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_s6755466524823107622T_VEBT(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5852_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_real,F2: fun_real_real,Bound: real,I: real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F2,X3)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F2,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_real(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5853_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_o,F2: fun_o_real,Bound: real,I: real] :
      ( ! [X3: $o] :
          ( aa_set_o_o(member_o((X3)),set_o2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_o_real(F2,(X3))),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_o_real(F2,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_o(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5854_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_nat,F2: fun_nat_real,Bound: real,I: real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F2,X3)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F2,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_size_list_nat(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5855_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs: list_VEBT_VEBTi,F2: fun_VEBT_VEBTi_real,Bound: real,I: real] :
      ( ! [X3: vEBT_VEBTi] :
          ( member_VEBT_VEBTi(X3,set_VEBT_VEBTi2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBTi_real(F2,X3)),Bound) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBTi_real(F2,Xs)),I)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,size_s7982070591426661849_VEBTi(Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_5856_real__nat__list,axiom,
    ! [F2: fun_VEBT_VEBT_nat,Xs: list_VEBT_VEBT,C2: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(F2,Xs)),C2)) = aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(aTP_Lamm_ea(F2),Xs)),aa_nat_real(semiri5074537144036343181t_real,C2)) ).

% real_nat_list
tff(fact_5857_f__g__map__foldr__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F2: fun_VEBT_VEBT_real,C2: real,G: fun_VEBT_VEBT_real,D2: real] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_VEBT_VEBT_real(F2,X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_VEBT_VEBT_real(G,X3))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F2,Xs)),D2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(G,Xs)),zero_zero_real))),D2)) ) ).

% f_g_map_foldr_bound
tff(fact_5858_f__g__map__foldr__bound,axiom,
    ! [Xs: list_nat,F2: fun_nat_real,C2: real,G: fun_nat_real,D2: real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F2,X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_nat_real(G,X3))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F2,Xs)),D2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(G,Xs)),zero_zero_real))),D2)) ) ).

% f_g_map_foldr_bound
tff(fact_5859_f__g__map__foldr__bound,axiom,
    ! [Xs: list_real,F2: fun_real_real,C2: real,G: fun_real_real,D2: real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F2,X3)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_real_real(G,X3))) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F2,Xs)),D2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(G,Xs)),zero_zero_real))),D2)) ) ).

% f_g_map_foldr_bound
tff(fact_5860_VEBTi_Osize_I3_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12: nat,X13a: array_VEBT_VEBTi,X14a: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11a,X12,X13a),X14a)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(size_size_VEBT_VEBTi,X13a)),aa_VEBT_VEBTi_nat(size_size_VEBT_VEBTi,X14a))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size(3)
tff(fact_5861_vebt__memberi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xc: nat,Ta: vEBT_VEBT] : refine_Imp_refines_o(vEBT_vebt_memberi(Tib,Xc),vEBT_V854960066525838166emberi(Ta,Tib,Xc)) ).

% vebt_memberi_refines
tff(fact_5862_refines__replicate,axiom,
    ! [F2: heap_Time_Heap_o,F5: heap_Time_Heap_o,Nb: nat] :
      ( refine_Imp_refines_o(F2,F5)
     => refine5896690332125372649list_o(vEBT_V2326993469660664182atei_o(Nb,F2),vEBT_V2326993469660664182atei_o(Nb,F5)) ) ).

% refines_replicate
tff(fact_5863_refines__replicate,axiom,
    ! [F2: heap_T2636463487746394924on_nat,F5: heap_T2636463487746394924on_nat,Nb: nat] :
      ( refine7594492741263601813on_nat(F2,F5)
     => refine1935026298455697829on_nat(vEBT_V792416675989592002on_nat(Nb,F2),vEBT_V792416675989592002on_nat(Nb,F5)) ) ).

% refines_replicate
tff(fact_5864_refines__replicate,axiom,
    ! [F2: heap_T8145700208782473153_VEBTi,F5: heap_T8145700208782473153_VEBTi,Nb: nat] :
      ( refine5565527176597971370_VEBTi(F2,F5)
     => refine3700189196150522554_VEBTi(vEBT_V1859673955506687831_VEBTi(Nb,F2),vEBT_V1859673955506687831_VEBTi(Nb,F5)) ) ).

% refines_replicate
tff(fact_5865_listsum__bound,axiom,
    ! [Xs: list_int,F2: fun_int_real,Ya: real] :
      ( ! [X3: int] :
          ( aa_set_int_o(member_int(X3),set_int2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F2,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(foldr_real_real(plus_plus_real,map_int_real(F2,Xs)),Ya)) ) ).

% listsum_bound
tff(fact_5866_listsum__bound,axiom,
    ! [Xs: list_complex,F2: fun_complex_real,Ya: real] :
      ( ! [X3: complex] :
          ( aa_set_complex_o(member_complex(X3),set_complex2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F2,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(foldr_real_real(plus_plus_real,map_complex_real(F2,Xs)),Ya)) ) ).

% listsum_bound
tff(fact_5867_listsum__bound,axiom,
    ! [Xs: list_VEBT_VEBT,F2: fun_VEBT_VEBT_real,Ya: real] :
      ( ! [X3: vEBT_VEBT] :
          ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_VEBT_VEBT_real(F2,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(F2,Xs)),Ya)) ) ).

% listsum_bound
tff(fact_5868_listsum__bound,axiom,
    ! [Xs: list_nat,F2: fun_nat_real,Ya: real] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),set_nat2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F2,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(foldr_real_real(plus_plus_real,map_nat_real(F2,Xs)),Ya)) ) ).

% listsum_bound
tff(fact_5869_listsum__bound,axiom,
    ! [Xs: list_real,F2: fun_real_real,Ya: real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_real2(Xs))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F2,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(foldr_real_real(plus_plus_real,map_real_real(F2,Xs)),Ya)) ) ).

% listsum_bound
tff(fact_5870_refines__case__VEBTi,axiom,
    ! [Tib: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_1207901889202581752Heap_o,F12: fun_o_1207901889202581752Heap_o,F22: fun_op1763656958002597022Heap_o,F23: fun_op1763656958002597022Heap_o] :
      ( ( Tib = Ti )
     => ( ! [A3: $o,B3: $o] : refine_Imp_refines_o(aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F1,(A3)),(B3)),aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F12,(A3)),(B3)))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : refine_Imp_refines_o(aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F22,Info2),Deg2),TreeArray),Summary),aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F23,Info2),Deg2),TreeArray),Summary))
         => refine_Imp_refines_o(vEBT_c6104975476656191286Heap_o(F22,F1,Tib),vEBT_c6104975476656191286Heap_o(F23,F12,Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5871_refines__case__VEBTi,axiom,
    ! [Tib: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_4049874799390722306on_nat,F12: fun_o_4049874799390722306on_nat,F22: fun_op5783775701855621896on_nat,F23: fun_op5783775701855621896on_nat] :
      ( ( Tib = Ti )
     => ( ! [A3: $o,B3: $o] : refine7594492741263601813on_nat(aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F1,(A3)),(B3)),aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F12,(A3)),(B3)))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : refine7594492741263601813on_nat(aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F22,Info2),Deg2),TreeArray),Summary),aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F23,Info2),Deg2),TreeArray),Summary))
         => refine7594492741263601813on_nat(vEBT_c6250501799366334488on_nat(F22,F1,Tib),vEBT_c6250501799366334488on_nat(F23,F12,Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5872_refines__case__VEBTi,axiom,
    ! [Tib: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun_o_6972675622308511895_VEBTi,F12: fun_o_6972675622308511895_VEBTi,F22: fun_op5127434927081926045_VEBTi,F23: fun_op5127434927081926045_VEBTi] :
      ( ( Tib = Ti )
     => ( ! [A3: $o,B3: $o] : refine5565527176597971370_VEBTi(aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F1,(A3)),(B3)),aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F12,(A3)),(B3)))
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : refine5565527176597971370_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F22,Info2),Deg2),TreeArray),Summary),aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F23,Info2),Deg2),TreeArray),Summary))
         => refine5565527176597971370_VEBTi(vEBT_c6028912655521741485_VEBTi(F22,F1,Tib),vEBT_c6028912655521741485_VEBTi(F23,F12,Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_5873_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_real(vEBT_VEBT_cnt,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info,Deg,TreeLista),Summarya)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summarya))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeLista)),zero_zero_real)) ).

% VEBT_internal.cnt.simps(2)
tff(fact_5874_VEBT__internal_Ocnt_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Xc) = Ya )
     => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( Ya != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary) )
             => ( Ya != aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_5875_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12: nat,X13a: array_VEBT_VEBTi,X14a: vEBT_VEBTi] : aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(X11a,X12,X13a),X14a)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_a6397454172108246045_VEBTi(vEBT_size_VEBTi,X13a)),aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,X14a))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBTi.size_gen(1)
tff(fact_5876_round__unique,axiom,
    ! [Xc: real,Ya: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Ya))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Ya)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xc) = Ya ) ) ) ).

% round_unique
tff(fact_5877_round__unique,axiom,
    ! [Xc: rat,Ya: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xc),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Ya))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Ya)),aa_rat_rat(plus_plus_rat(Xc),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,Xc) = Ya ) ) ) ).

% round_unique
tff(fact_5878_mult__le__cancel__iff2,axiom,
    ! [Z: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ) ).

% mult_le_cancel_iff2
tff(fact_5879_mult__le__cancel__iff2,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Z),Xc)),aa_rat_rat(times_times_rat(Z),Ya))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya) ) ) ).

% mult_le_cancel_iff2
tff(fact_5880_mult__le__cancel__iff2,axiom,
    ! [Z: int,Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Z),Xc)),aa_int_int(times_times_int(Z),Ya))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya) ) ) ).

% mult_le_cancel_iff2
tff(fact_5881_mult__le__cancel__iff1,axiom,
    ! [Z: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ) ).

% mult_le_cancel_iff1
tff(fact_5882_mult__le__cancel__iff1,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(Xc),Z)),aa_rat_rat(times_times_rat(Ya),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya) ) ) ).

% mult_le_cancel_iff1
tff(fact_5883_mult__le__cancel__iff1,axiom,
    ! [Z: int,Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(times_times_int(Xc),Z)),aa_int_int(times_times_int(Ya),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),Ya) ) ) ).

% mult_le_cancel_iff1
tff(fact_5884_vebt__predi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xc: nat,Ta: vEBT_VEBT] : refine7594492741263601813on_nat(vEBT_vebt_predi(Tib,Xc),vEBT_VEBT_vebt_predi(Ta,Tib,Xc)) ).

% vebt_predi_refines
tff(fact_5885_vebt__succi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xc: nat,Ta: vEBT_VEBT] : refine7594492741263601813on_nat(vEBT_vebt_succi(Tib,Xc),vEBT_VEBT_vebt_succi(Ta,Tib,Xc)) ).

% vebt_succi_refines
tff(fact_5886_vebt__buildupi__refines,axiom,
    ! [Nb: nat] : refine5565527176597971370_VEBTi(vEBT_vebt_buildupi(Nb),vEBT_V739175172307565963ildupi(Nb)) ).

% vebt_buildupi_refines
tff(fact_5887_vebt__inserti__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xc: nat,Ta: vEBT_VEBT] : refine5565527176597971370_VEBTi(vEBT_vebt_inserti(Tib,Xc),vEBT_V3964819847710782039nserti(Ta,Tib,Xc)) ).

% vebt_inserti_refines
tff(fact_5888_round__0,axiom,
    aa_real_int(archim8280529875227126926d_real,zero_zero_real) = zero_zero_int ).

% round_0
tff(fact_5889_round__0,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,zero_zero_rat) = zero_zero_int ).

% round_0
tff(fact_5890_round__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_5891_round__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% round_numeral
tff(fact_5892_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,F2: fun_o_6972675622308511895_VEBTi,Bnd: fun_o_fun_o_nat,F5: fun_op5127434927081926045_VEBTi,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: $o,B3: $o] :
          ( ( Tib = aa_o_VEBT_VEBTi(vEBT_Leafi((A3)),(B3)) )
         => aa_nat_o(time_T5737551269749752165_VEBTi(aa_o_H7577119287286566245_VEBTi(aa_o_f5262348187908705736_VEBTi(F2,(A3)),(B3))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A3)),(B3))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary) )
           => aa_nat_o(time_T5737551269749752165_VEBTi(aa_VEB836174949890266638_VEBTi(aa_arr2912372873376718191_VEBTi(aa_nat6445230840460233604_VEBTi(aa_opt5182041009420754510_VEBTi(F5,Info2),Deg2),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary)) )
       => aa_nat_o(time_T5737551269749752165_VEBTi(vEBT_c6028912655521741485_VEBTi(F5,F2,Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_5893_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,F2: fun_o_1207901889202581752Heap_o,Bnd: fun_o_fun_o_nat,F5: fun_op1763656958002597022Heap_o,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: $o,B3: $o] :
          ( ( Tib = aa_o_VEBT_VEBTi(vEBT_Leafi((A3)),(B3)) )
         => aa_nat_o(time_TBOUND_o(aa_o_H7399033255965945470Heap_o(aa_o_f869145629572826609Heap_o(F2,(A3)),(B3))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A3)),(B3))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary) )
           => aa_nat_o(time_TBOUND_o(aa_VEB8194921341634505877Heap_o(aa_arr2373296168907467658Heap_o(aa_nat8650107858160254923Heap_o(aa_opt6539295502271640599Heap_o(F5,Info2),Deg2),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary)) )
       => aa_nat_o(time_TBOUND_o(vEBT_c6104975476656191286Heap_o(F5,F2,Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_5894_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,F2: fun_o_4049874799390722306on_nat,Bnd: fun_o_fun_o_nat,F5: fun_op5783775701855621896on_nat,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: $o,B3: $o] :
          ( ( Tib = aa_o_VEBT_VEBTi(vEBT_Leafi((A3)),(B3)) )
         => aa_nat_o(time_T8353473612707095248on_nat(aa_o_H4215342640551218896on_nat(aa_o_f5606600903069576627on_nat(F2,(A3)),(B3))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A3)),(B3))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary) )
           => aa_nat_o(time_T8353473612707095248on_nat(aa_VEB2711320435582112505on_nat(aa_arr2955191963898321370on_nat(aa_nat5317369504323972463on_nat(aa_opt8624683241416710969on_nat(F5,Info2),Deg2),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary)) )
       => aa_nat_o(time_T8353473612707095248on_nat(vEBT_c6250501799366334488on_nat(F5,F2,Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_5895_TBOUND__VEBT__case,axiom,
    ! [Tib: vEBT_VEBTi,F2: fun_o_7009304396571837746ap_nat,Bnd: fun_o_fun_o_nat,F5: fun_op2425996437410981688ap_nat,Bnd2: fun_op7256438278201533260Ti_nat] :
      ( ! [A3: $o,B3: $o] :
          ( ( Tib = aa_o_VEBT_VEBTi(vEBT_Leafi((A3)),(B3)) )
         => aa_nat_o(time_TBOUND_nat(aa_o_H8739796202006397184ap_nat(aa_o_f2078835126877633507ap_nat(F2,(A3)),(B3))),aa_o_nat(aa_o_fun_o_nat(Bnd,(A3)),(B3))) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Tib = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Info2,Deg2,TreeArray),Summary) )
           => aa_nat_o(time_TBOUND_nat(aa_VEB183921879236458153ap_nat(aa_arr6934346397072338570ap_nat(aa_nat1288005622339309855ap_nat(aa_opt6697879614838309225ap_nat(F5,Info2),Deg2),TreeArray),Summary)),aa_VEBT_VEBTi_nat(aa_arr8157624991474932664Ti_nat(aa_nat7496786144979625977Ti_nat(aa_opt8327319410639121349Ti_nat(Bnd2,Info2),Deg2),TreeArray),Summary)) )
       => aa_nat_o(time_TBOUND_nat(vEBT_c1335663792808957512ap_nat(F5,F2,Tib)),aa_VEBT_VEBTi_nat(vEBT_case_VEBTi_nat(Bnd2,Bnd),Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_5896_round__mono,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,Xc)),aa_rat_int(archim7778729529865785530nd_rat,Ya)) ) ).

% round_mono
tff(fact_5897_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBTi_nat(vEBT_size_VEBTi,aa_o_VEBT_VEBTi(vEBT_Leafi((X21)),(X222))) = zero_zero_nat ).

% VEBTi.size_gen(2)
tff(fact_5898_mult__less__iff1,axiom,
    ! [Z: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Z)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ) ).

% mult_less_iff1
tff(fact_5899_mult__less__iff1,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(Xc),Z)),aa_rat_rat(times_times_rat(Ya),Z))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya) ) ) ).

% mult_less_iff1
tff(fact_5900_mult__less__iff1,axiom,
    ! [Z: int,Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(Xc),Z)),aa_int_int(times_times_int(Ya),Z))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Ya) ) ) ).

% mult_less_iff1
tff(fact_5901_of__int__round__le,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_5902_of__int__round__le,axiom,
    ! [Xc: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xc))),aa_rat_rat(plus_plus_rat(Xc),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_5903_of__int__round__ge,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xc))) ).

% of_int_round_ge
tff(fact_5904_of__int__round__ge,axiom,
    ! [Xc: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Xc),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xc))) ).

% of_int_round_ge
tff(fact_5905_of__int__round__gt,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xc))) ).

% of_int_round_gt
tff(fact_5906_of__int__round__gt,axiom,
    ! [Xc: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xc),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xc))) ).

% of_int_round_gt
tff(fact_5907_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5908_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Nb),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),unique5024387138958732305ep_int(aa_num_num(bit1,Nb),unique5052692396658037445od_int(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5909_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,M),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,M))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_5910_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5911_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),unique5024387138958732305ep_int(aa_num_num(bit1,Nb),unique5052692396658037445od_int(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5912_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,M))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,M),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_5913_divides__aux__eq,axiom,
    ! [Q3: code_integer,R3: code_integer] :
      ( unique5706413561485394159nteger(produc1086072967326762835nteger(Q3,R3))
    <=> ( R3 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
tff(fact_5914_divides__aux__eq,axiom,
    ! [Q3: nat,R3: nat] :
      ( unique6322359934112328802ux_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q3),R3))
    <=> ( R3 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_5915_divides__aux__eq,axiom,
    ! [Q3: int,R3: int] :
      ( aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,product_Pair_int_int(Q3,R3))
    <=> ( R3 = zero_zero_int ) ) ).

% divides_aux_eq
tff(fact_5916_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [Xc: nat,Ya: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,Xc) = Ya )
     => ( ( ( Xc = zero_zero_nat )
         => ( Ya != one_one_int ) )
       => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
           => ( Ya != one_one_int ) )
         => ~ ! [N: nat] :
                ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Ya != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_5917_arsinh__0,axiom,
    aa_real_real(arsinh_real,zero_zero_real) = zero_zero_real ).

% arsinh_0
tff(fact_5918_artanh__0,axiom,
    aa_real_real(artanh_real,zero_zero_real) = zero_zero_real ).

% artanh_0
tff(fact_5919_dvd__0__left__iff,axiom,
    ! [A2: complex] :
      ( aa_complex_o(dvd_dvd_complex(zero_zero_complex),A2)
    <=> ( A2 = zero_zero_complex ) ) ).

% dvd_0_left_iff
tff(fact_5920_dvd__0__left__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A2)
    <=> ( A2 = zero_zero_real ) ) ).

% dvd_0_left_iff
tff(fact_5921_dvd__0__left__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A2)
    <=> ( A2 = zero_zero_rat ) ) ).

% dvd_0_left_iff
tff(fact_5922_dvd__0__left__iff,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A2)
    <=> ( A2 = zero_zero_nat ) ) ).

% dvd_0_left_iff
tff(fact_5923_dvd__0__left__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A2)
    <=> ( A2 = zero_zero_int ) ) ).

% dvd_0_left_iff
tff(fact_5924_dvd__0__right,axiom,
    ! [A2: complex] : aa_complex_o(dvd_dvd_complex(A2),zero_zero_complex) ).

% dvd_0_right
tff(fact_5925_dvd__0__right,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),zero_zero_real) ).

% dvd_0_right
tff(fact_5926_dvd__0__right,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),zero_zero_rat) ).

% dvd_0_right
tff(fact_5927_dvd__0__right,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),zero_zero_nat) ).

% dvd_0_right
tff(fact_5928_dvd__0__right,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),zero_zero_int) ).

% dvd_0_right
tff(fact_5929_dvd__add__triv__right__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A2))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_5930_dvd__add__triv__right__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(B2),A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_5931_dvd__add__triv__right__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_5932_dvd__add__triv__right__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(plus_plus_int(B2),A2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_5933_dvd__add__triv__left__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_5934_dvd__add__triv__left__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(A2),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_5935_dvd__add__triv__left__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_5936_dvd__add__triv__left__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(plus_plus_int(A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_5937_div__dvd__div,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),A2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C2) ) ) ) ).

% div_dvd_div
tff(fact_5938_div__dvd__div,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),A2))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C2) ) ) ) ).

% div_dvd_div
tff(fact_5939_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),one_one_nat)
    <=> ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_5940_dvd__times__right__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C2) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5941_dvd__times__right__cancel__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(B2),A2)),aa_int_int(times_times_int(C2),A2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C2) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5942_dvd__times__left__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C2) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5943_dvd__times__left__cancel__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A2),B2)),aa_int_int(times_times_int(A2),C2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C2) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5944_dvd__mult__cancel__right,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(A2),C2)),aa_complex_complex(times_times_complex(B2),C2))
    <=> ( ( C2 = zero_zero_complex )
        | aa_complex_o(dvd_dvd_complex(A2),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_5945_dvd__mult__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))
    <=> ( ( C2 = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_5946_dvd__mult__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),C2))
    <=> ( ( C2 = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_5947_dvd__mult__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2))
    <=> ( ( C2 = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_5948_dvd__mult__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(times_times_complex(C2),A2)),aa_complex_complex(times_times_complex(C2),B2))
    <=> ( ( C2 = zero_zero_complex )
        | aa_complex_o(dvd_dvd_complex(A2),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_5949_dvd__mult__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2))
    <=> ( ( C2 = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_5950_dvd__mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(C2),A2)),aa_rat_rat(times_times_rat(C2),B2))
    <=> ( ( C2 = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_5951_dvd__mult__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2))
    <=> ( ( C2 = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_5952_dvd__add__times__triv__right__iff,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5953_dvd__add__times__triv__right__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(B2),aa_rat_rat(times_times_rat(C2),A2)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5954_dvd__add__times__triv__right__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5955_dvd__add__times__triv__right__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(plus_plus_int(B2),aa_int_int(times_times_int(C2),A2)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_5956_dvd__add__times__triv__left__iff,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),A2)),B2))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5957_dvd__add__times__triv__left__iff,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(aa_rat_rat(times_times_rat(C2),A2)),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5958_dvd__add__times__triv__left__iff,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5959_dvd__add__times__triv__left__iff,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(plus_plus_int(aa_int_int(times_times_int(C2),A2)),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_5960_unit__prod,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),one_one_nat) ) ) ).

% unit_prod
tff(fact_5961_unit__prod,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A2),B2)),one_one_int) ) ) ).

% unit_prod
tff(fact_5962_div__add,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) ) ) ) ).

% div_add
tff(fact_5963_div__add,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A2),B2)),C2) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) ) ) ) ).

% div_add
tff(fact_5964_unit__div__1__div__1,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2)) = A2 ) ) ).

% unit_div_1_div_1
tff(fact_5965_unit__div__1__div__1,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2)) = A2 ) ) ).

% unit_div_1_div_1
tff(fact_5966_unit__div__1__unit,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_5967_unit__div__1__unit,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_5968_unit__div,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),one_one_nat) ) ) ).

% unit_div
tff(fact_5969_unit__div,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),one_one_int) ) ) ).

% unit_div
tff(fact_5970_dvd__mult__div__cancel,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_5971_dvd__mult__div__cancel,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_int(times_times_int(A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_5972_dvd__div__mult__self,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)),A2) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_5973_dvd__div__mult__self,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)),A2) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_5974_div__diff,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(A2),B2)),C2) = aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) ) ) ) ).

% div_diff
tff(fact_5975_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(suc,zero_zero_nat))
    <=> ( M = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_5976_dvd__1__left,axiom,
    ! [K: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(suc,zero_zero_nat)),K) ).

% dvd_1_left
tff(fact_5977_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
    <=> ( ( K = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_5978_unit__div__mult__self,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2)),A2) = B2 ) ) ).

% unit_div_mult_self
tff(fact_5979_unit__div__mult__self,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2)),A2) = B2 ) ) ).

% unit_div_mult_self
tff(fact_5980_unit__mult__div__div,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) ) ) ).

% unit_mult_div_div
tff(fact_5981_unit__mult__div__div,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_int(times_times_int(B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) ) ) ).

% unit_mult_div_div
tff(fact_5982_even__add,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ) ) ).

% even_add
tff(fact_5983_even__add,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(A2),B2))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2) ) ) ).

% even_add
tff(fact_5984_odd__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))
    <=> ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ) ) ).

% odd_add
tff(fact_5985_odd__add,axiom,
    ! [A2: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(A2),B2))
    <=> ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2) ) ) ).

% odd_add
tff(fact_5986_even__mult__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) ) ) ).

% even_mult_iff
tff(fact_5987_even__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(times_times_int(A2),B2))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2) ) ) ).

% even_mult_iff
tff(fact_5988_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_5989_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_Suc
tff(fact_5990_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5055182867167087721od_nat(M,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,M)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_5991_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique5052692396658037445od_int(M,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,M),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_5992_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] : unique3479559517661332726nteger(M,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(M),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_5993_dvd__numeral__simp,axiom,
    ! [M: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> unique6322359934112328802ux_nat(unique5055182867167087721od_nat(Nb,M)) ) ).

% dvd_numeral_simp
tff(fact_5994_dvd__numeral__simp,axiom,
    ! [M: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_Pro4592068669816069595_int_o(unique6319869463603278526ux_int,unique5052692396658037445od_int(Nb,M)) ) ).

% dvd_numeral_simp
tff(fact_5995_dvd__numeral__simp,axiom,
    ! [M: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Nb))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(Nb,M)) ) ).

% dvd_numeral_simp
tff(fact_5996_even__plus__one__iff,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2) ) ).

% even_plus_one_iff
tff(fact_5997_even__plus__one__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(A2),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2) ) ).

% even_plus_one_iff
tff(fact_5998_even__diff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(minus_minus_int(A2),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(A2),B2)) ) ).

% even_diff
tff(fact_5999_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_Suc_div_two
tff(fact_6000_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% odd_Suc_div_two
tff(fact_6001_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit0,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(3)
tff(fact_6002_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit0,Nb)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(3)
tff(fact_6003_divmod__algorithm__code_I3_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit0,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(3)
tff(fact_6004_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5055182867167087721od_nat(one,aa_num_num(bit1,Nb)) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,one)) ).

% divmod_algorithm_code(4)
tff(fact_6005_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique5052692396658037445od_int(one,aa_num_num(bit1,Nb)) = product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,one)) ).

% divmod_algorithm_code(4)
tff(fact_6006_divmod__algorithm__code_I4_J,axiom,
    ! [Nb: num] : unique3479559517661332726nteger(one,aa_num_num(bit1,Nb)) = produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(one)) ).

% divmod_algorithm_code(4)
tff(fact_6007_even__succ__div__two,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_6008_even__succ__div__two,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A2),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_6009_odd__succ__div__two,axiom,
    ! [A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_6010_odd__succ__div__two,axiom,
    ! [A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A2),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_6011_even__succ__div__2,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_6012_even__succ__div__2,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(one_one_int),A2)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_6013_even__power,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),power_8256067586552552935nteger(A2,Nb))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_6014_even__power,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_6015_even__power,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(A2),Nb))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% even_power
tff(fact_6016_zero__le__power__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6017_zero__le__power__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6018_zero__le__power__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6019_zero__le__power__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_6020_power__less__zero__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6021_power__less__zero__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6022_power__less__zero__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6023_power__less__zero__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ) ).

% power_less_zero_eq_numeral
tff(fact_6024_power__less__zero__eq,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(A2,Nb)),zero_z3403309356797280102nteger)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger) ) ) ).

% power_less_zero_eq
tff(fact_6025_power__less__zero__eq,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),zero_zero_real)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ) ).

% power_less_zero_eq
tff(fact_6026_power__less__zero__eq,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A2),Nb)),zero_zero_rat)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ) ).

% power_less_zero_eq
tff(fact_6027_power__less__zero__eq,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A2),Nb)),zero_zero_int)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ) ).

% power_less_zero_eq
tff(fact_6028_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6029_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6030_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_6031_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_6032_even__diff__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)) ) ) ).

% even_diff_nat
tff(fact_6033_odd__two__times__div__two__succ,axiom,
    ! [A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_nat) = A2 ) ) ).

% odd_two_times_div_two_succ
tff(fact_6034_odd__two__times__div__two__succ,axiom,
    ! [A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A2 ) ) ).

% odd_two_times_div_two_succ
tff(fact_6035_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb),one_one_Code_integer))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6036_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6037_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_6038_zero__less__power__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A2 != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6039_zero__less__power__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A2 != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6040_zero__less__power__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A2 != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6041_zero__less__power__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,W)))
    <=> ( ( aa_num_nat(numeral_numeral_nat,W) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & ( A2 != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_6042_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat) ) ) ).

% odd_two_times_div_two_nat
tff(fact_6043_power__le__zero__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A2 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6044_power__le__zero__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,W))),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A2 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6045_power__le__zero__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A2 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6046_power__le__zero__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,W))),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_num_nat(numeral_numeral_nat,W))
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
            & ( A2 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_6047_even__succ__div__exp,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A2),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = divide6298287555418463151nteger(A2,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_6048_even__succ__div__exp,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_6049_even__succ__div__exp,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(one_one_int),A2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_6050_dvd__antisym,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M)
       => ( M = Nb ) ) ) ).

% dvd_antisym
tff(fact_6051_dvd__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2) ) ) ).

% dvd_trans
tff(fact_6052_dvd__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2) ) ) ).

% dvd_trans
tff(fact_6053_dvd__refl,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),A2) ).

% dvd_refl
tff(fact_6054_dvd__refl,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),A2) ).

% dvd_refl
tff(fact_6055_of__nat__dvd__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6056_of__nat__dvd__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6057_of__nat__dvd__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ).

% of_nat_dvd_iff
tff(fact_6058_dvd__0__left,axiom,
    ! [A2: complex] :
      ( aa_complex_o(dvd_dvd_complex(zero_zero_complex),A2)
     => ( A2 = zero_zero_complex ) ) ).

% dvd_0_left
tff(fact_6059_dvd__0__left,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,zero_zero_real),A2)
     => ( A2 = zero_zero_real ) ) ).

% dvd_0_left
tff(fact_6060_dvd__0__left,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,zero_zero_rat),A2)
     => ( A2 = zero_zero_rat ) ) ).

% dvd_0_left
tff(fact_6061_dvd__0__left,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A2)
     => ( A2 = zero_zero_nat ) ) ).

% dvd_0_left
tff(fact_6062_dvd__0__left,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),A2)
     => ( A2 = zero_zero_int ) ) ).

% dvd_0_left
tff(fact_6063_dvd__field__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_complex_o(dvd_dvd_complex(A2),B2)
    <=> ( ( A2 = zero_zero_complex )
       => ( B2 = zero_zero_complex ) ) ) ).

% dvd_field_iff
tff(fact_6064_dvd__field__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2)
    <=> ( ( A2 = zero_zero_real )
       => ( B2 = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_6065_dvd__field__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2)
    <=> ( ( A2 = zero_zero_rat )
       => ( B2 = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_6066_dvd__add__right__iff,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),C2) ) ) ).

% dvd_add_right_iff
tff(fact_6067_dvd__add__right__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(B2),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C2) ) ) ).

% dvd_add_right_iff
tff(fact_6068_dvd__add__right__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2) ) ) ).

% dvd_add_right_iff
tff(fact_6069_dvd__add__right__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(plus_plus_int(B2),C2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2) ) ) ).

% dvd_add_right_iff
tff(fact_6070_dvd__add__left__iff,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),C2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2) ) ) ).

% dvd_add_left_iff
tff(fact_6071_dvd__add__left__iff,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(B2),C2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2) ) ) ).

% dvd_add_left_iff
tff(fact_6072_dvd__add__left__iff,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ) ).

% dvd_add_left_iff
tff(fact_6073_dvd__add__left__iff,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(plus_plus_int(B2),C2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ) ).

% dvd_add_left_iff
tff(fact_6074_dvd__add,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),C2)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C2)) ) ) ).

% dvd_add
tff(fact_6075_dvd__add,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C2)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(plus_plus_rat(B2),C2)) ) ) ).

% dvd_add
tff(fact_6076_dvd__add,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C2)) ) ) ).

% dvd_add
tff(fact_6077_dvd__add,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(plus_plus_int(B2),C2)) ) ) ).

% dvd_add
tff(fact_6078_dvd__unit__imp__unit,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_6079_dvd__unit__imp__unit,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_6080_unit__imp__dvd,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2) ) ).

% unit_imp_dvd
tff(fact_6081_unit__imp__dvd,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2) ) ).

% unit_imp_dvd
tff(fact_6082_one__dvd,axiom,
    ! [A2: assn] : dvd_dvd_assn(one_one_assn,A2) ).

% one_dvd
tff(fact_6083_one__dvd,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A2) ).

% one_dvd
tff(fact_6084_one__dvd,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A2) ).

% one_dvd
tff(fact_6085_one__dvd,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A2) ).

% one_dvd
tff(fact_6086_one__dvd,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A2) ).

% one_dvd
tff(fact_6087_dvd__triv__right,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A2)) ).

% dvd_triv_right
tff(fact_6088_dvd__triv__right,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(B2),A2)) ).

% dvd_triv_right
tff(fact_6089_dvd__triv__right,axiom,
    ! [A2: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)) ).

% dvd_triv_right
tff(fact_6090_dvd__triv__right,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(times_times_int(B2),A2)) ).

% dvd_triv_right
tff(fact_6091_dvd__triv__right,axiom,
    ! [A2: assn,B2: assn] : dvd_dvd_assn(A2,aa_assn_assn(times_times_assn(B2),A2)) ).

% dvd_triv_right
tff(fact_6092_dvd__mult__right,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),C2)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),C2) ) ).

% dvd_mult_right
tff(fact_6093_dvd__mult__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),B2)),C2)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),C2) ) ).

% dvd_mult_right
tff(fact_6094_dvd__mult__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C2) ) ).

% dvd_mult_right
tff(fact_6095_dvd__mult__right,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A2),B2)),C2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C2) ) ).

% dvd_mult_right
tff(fact_6096_dvd__mult__right,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( dvd_dvd_assn(aa_assn_assn(times_times_assn(A2),B2),C2)
     => dvd_dvd_assn(B2,C2) ) ).

% dvd_mult_right
tff(fact_6097_mult__dvd__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C2),D2)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2)) ) ) ).

% mult_dvd_mono
tff(fact_6098_mult__dvd__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C2),D2)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(times_times_rat(B2),D2)) ) ) ).

% mult_dvd_mono
tff(fact_6099_mult__dvd__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2)) ) ) ).

% mult_dvd_mono
tff(fact_6100_mult__dvd__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),D2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),D2)) ) ) ).

% mult_dvd_mono
tff(fact_6101_mult__dvd__mono,axiom,
    ! [A2: assn,B2: assn,C2: assn,D2: assn] :
      ( dvd_dvd_assn(A2,B2)
     => ( dvd_dvd_assn(C2,D2)
       => dvd_dvd_assn(aa_assn_assn(times_times_assn(A2),C2),aa_assn_assn(times_times_assn(B2),D2)) ) ) ).

% mult_dvd_mono
tff(fact_6102_dvd__triv__left,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) ).

% dvd_triv_left
tff(fact_6103_dvd__triv__left,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(A2),B2)) ).

% dvd_triv_left
tff(fact_6104_dvd__triv__left,axiom,
    ! [A2: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)) ).

% dvd_triv_left
tff(fact_6105_dvd__triv__left,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(times_times_int(A2),B2)) ).

% dvd_triv_left
tff(fact_6106_dvd__triv__left,axiom,
    ! [A2: assn,B2: assn] : dvd_dvd_assn(A2,aa_assn_assn(times_times_assn(A2),B2)) ).

% dvd_triv_left
tff(fact_6107_dvd__mult__left,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)),C2)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),C2) ) ).

% dvd_mult_left
tff(fact_6108_dvd__mult__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(times_times_rat(A2),B2)),C2)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C2) ) ).

% dvd_mult_left
tff(fact_6109_dvd__mult__left,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2) ) ).

% dvd_mult_left
tff(fact_6110_dvd__mult__left,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A2),B2)),C2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2) ) ).

% dvd_mult_left
tff(fact_6111_dvd__mult__left,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( dvd_dvd_assn(aa_assn_assn(times_times_assn(A2),B2),C2)
     => dvd_dvd_assn(A2,C2) ) ).

% dvd_mult_left
tff(fact_6112_dvd__mult2,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ) ).

% dvd_mult2
tff(fact_6113_dvd__mult2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(B2),C2)) ) ).

% dvd_mult2
tff(fact_6114_dvd__mult2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ) ).

% dvd_mult2
tff(fact_6115_dvd__mult2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(times_times_int(B2),C2)) ) ).

% dvd_mult2
tff(fact_6116_dvd__mult2,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( dvd_dvd_assn(A2,B2)
     => dvd_dvd_assn(A2,aa_assn_assn(times_times_assn(B2),C2)) ) ).

% dvd_mult2
tff(fact_6117_dvd__mult,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),C2)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2)) ) ).

% dvd_mult
tff(fact_6118_dvd__mult,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),C2)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A2),aa_rat_rat(times_times_rat(B2),C2)) ) ).

% dvd_mult
tff(fact_6119_dvd__mult,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ) ).

% dvd_mult
tff(fact_6120_dvd__mult,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(times_times_int(B2),C2)) ) ).

% dvd_mult
tff(fact_6121_dvd__mult,axiom,
    ! [A2: assn,C2: assn,B2: assn] :
      ( dvd_dvd_assn(A2,C2)
     => dvd_dvd_assn(A2,aa_assn_assn(times_times_assn(B2),C2)) ) ).

% dvd_mult
tff(fact_6122_dvd__def,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2)
    <=> ? [K3: real] : A2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),K3) ) ).

% dvd_def
tff(fact_6123_dvd__def,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2)
    <=> ? [K3: rat] : A2 = aa_rat_rat(times_times_rat(B2),K3) ) ).

% dvd_def
tff(fact_6124_dvd__def,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
    <=> ? [K3: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K3) ) ).

% dvd_def
tff(fact_6125_dvd__def,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
    <=> ? [K3: int] : A2 = aa_int_int(times_times_int(B2),K3) ) ).

% dvd_def
tff(fact_6126_dvd__def,axiom,
    ! [B2: assn,A2: assn] :
      ( dvd_dvd_assn(B2,A2)
    <=> ? [K3: assn] : A2 = aa_assn_assn(times_times_assn(B2),K3) ) ).

% dvd_def
tff(fact_6127_dvdI,axiom,
    ! [A2: real,B2: real,K: real] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2) ) ).

% dvdI
tff(fact_6128_dvdI,axiom,
    ! [A2: rat,B2: rat,K: rat] :
      ( ( A2 = aa_rat_rat(times_times_rat(B2),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2) ) ).

% dvdI
tff(fact_6129_dvdI,axiom,
    ! [A2: nat,B2: nat,K: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2) ) ).

% dvdI
tff(fact_6130_dvdI,axiom,
    ! [A2: int,B2: int,K: int] :
      ( ( A2 = aa_int_int(times_times_int(B2),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2) ) ).

% dvdI
tff(fact_6131_dvdI,axiom,
    ! [A2: assn,B2: assn,K: assn] :
      ( ( A2 = aa_assn_assn(times_times_assn(B2),K) )
     => dvd_dvd_assn(B2,A2) ) ).

% dvdI
tff(fact_6132_dvdE,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2)
     => ~ ! [K2: real] : A2 != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),K2) ) ).

% dvdE
tff(fact_6133_dvdE,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2)
     => ~ ! [K2: rat] : A2 != aa_rat_rat(times_times_rat(B2),K2) ) ).

% dvdE
tff(fact_6134_dvdE,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
     => ~ ! [K2: nat] : A2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K2) ) ).

% dvdE
tff(fact_6135_dvdE,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ~ ! [K2: int] : A2 != aa_int_int(times_times_int(B2),K2) ) ).

% dvdE
tff(fact_6136_dvdE,axiom,
    ! [B2: assn,A2: assn] :
      ( dvd_dvd_assn(B2,A2)
     => ~ ! [K2: assn] : A2 != aa_assn_assn(times_times_assn(B2),K2) ) ).

% dvdE
tff(fact_6137_dvd__diff,axiom,
    ! [Xc: real,Ya: real,Z: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xc),Ya)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xc),Z)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),Z)) ) ) ).

% dvd_diff
tff(fact_6138_dvd__diff,axiom,
    ! [Xc: rat,Ya: rat,Z: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xc),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xc),Z)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xc),aa_rat_rat(minus_minus_rat(Ya),Z)) ) ) ).

% dvd_diff
tff(fact_6139_dvd__diff,axiom,
    ! [Xc: int,Ya: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xc),Ya)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xc),Z)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xc),aa_int_int(minus_minus_int(Ya),Z)) ) ) ).

% dvd_diff
tff(fact_6140_dvd__diff__commute,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(minus_minus_int(C2),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(minus_minus_int(B2),C2)) ) ).

% dvd_diff_commute
tff(fact_6141_div__div__div__same,axiom,
    ! [D2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),D2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),D2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ) ).

% div_div_div_same
tff(fact_6142_div__div__div__same,axiom,
    ! [D2: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),D2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),D2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ) ).

% div_div_div_same
tff(fact_6143_dvd__div__eq__cancel,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A2),C2) = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
     => ( aa_complex_o(dvd_dvd_complex(C2),A2)
       => ( aa_complex_o(dvd_dvd_complex(C2),B2)
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6144_dvd__div__eq__cancel,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C2),A2)
       => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C2),B2)
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6145_dvd__div__eq__cancel,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A2),C2) = aa_rat_rat(divide_divide_rat(B2),C2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C2),A2)
       => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C2),B2)
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6146_dvd__div__eq__cancel,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B2)
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6147_dvd__div__eq__cancel,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2) )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),A2)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_6148_dvd__div__eq__iff,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( aa_complex_o(dvd_dvd_complex(C2),A2)
     => ( aa_complex_o(dvd_dvd_complex(C2),B2)
       => ( ( aa_complex_complex(divide1717551699836669952omplex(A2),C2) = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6149_dvd__div__eq__iff,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C2),A2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,C2),B2)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),C2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6150_dvd__div__eq__iff,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C2),A2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C2),B2)
       => ( ( aa_rat_rat(divide_divide_rat(A2),C2) = aa_rat_rat(divide_divide_rat(B2),C2) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6151_dvd__div__eq__iff,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6152_dvd__div__eq__iff,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2) )
        <=> ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_6153_dvd__power__same,axiom,
    ! [Xc: nat,Ya: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xc),Ya)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ya),Nb)) ) ).

% dvd_power_same
tff(fact_6154_dvd__power__same,axiom,
    ! [Xc: real,Ya: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xc),Ya)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),Nb)) ) ).

% dvd_power_same
tff(fact_6155_dvd__power__same,axiom,
    ! [Xc: int,Ya: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xc),Ya)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xc),Nb)),aa_nat_int(power_power_int(Ya),Nb)) ) ).

% dvd_power_same
tff(fact_6156_dvd__power__same,axiom,
    ! [Xc: complex,Ya: complex,Nb: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xc),Ya)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(Xc),Nb)),aa_nat_complex(power_power_complex(Ya),Nb)) ) ).

% dvd_power_same
tff(fact_6157_dvd__power__same,axiom,
    ! [Xc: code_integer,Ya: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xc),Ya)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(Xc,Nb)),power_8256067586552552935nteger(Ya,Nb)) ) ).

% dvd_power_same
tff(fact_6158_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_6159_subset__divisors__dvd,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(collect_complex(aTP_Lamm_eb(A2))),collect_complex(aTP_Lamm_eb(B2)))
    <=> aa_complex_o(dvd_dvd_complex(A2),B2) ) ).

% subset_divisors_dvd
tff(fact_6160_subset__divisors__dvd,axiom,
    ! [A2: int,B2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ec,A2))),collect_int(aa_int_fun_int_o(aTP_Lamm_ec,B2)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ).

% subset_divisors_dvd
tff(fact_6161_subset__divisors__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ed,A2))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ed,B2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ).

% subset_divisors_dvd
tff(fact_6162_strict__subset__divisors__dvd,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_set_complex_o(ord_less_set_complex(collect_complex(aTP_Lamm_eb(A2))),collect_complex(aTP_Lamm_eb(B2)))
    <=> ( aa_complex_o(dvd_dvd_complex(A2),B2)
        & ~ aa_complex_o(dvd_dvd_complex(B2),A2) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6163_strict__subset__divisors__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_set_nat_o(ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ed,A2))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ed,B2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6164_strict__subset__divisors__dvd,axiom,
    ! [A2: int,B2: int] :
      ( aa_set_int_o(ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ec,A2))),collect_int(aa_int_fun_int_o(aTP_Lamm_ec,B2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2) ) ) ).

% strict_subset_divisors_dvd
tff(fact_6165_not__is__unit__0,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),one_one_nat) ).

% not_is_unit_0
tff(fact_6166_not__is__unit__0,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,zero_zero_int),one_one_int) ).

% not_is_unit_0
tff(fact_6167_pinf_I9_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S2)) ) ) ).

% pinf(9)
tff(fact_6168_pinf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X4),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X4),S2)) ) ) ).

% pinf(9)
tff(fact_6169_pinf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S2)) ) ) ).

% pinf(9)
tff(fact_6170_pinf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),S2)) ) ) ).

% pinf(9)
tff(fact_6171_pinf_I10_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X4)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S2))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S2)) ) ) ).

% pinf(10)
tff(fact_6172_pinf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X4)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X4),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X4),S2)) ) ) ).

% pinf(10)
tff(fact_6173_pinf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X4)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S2)) ) ) ).

% pinf(10)
tff(fact_6174_pinf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X4)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),S2)) ) ) ).

% pinf(10)
tff(fact_6175_minf_I9_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S2))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S2)) ) ) ).

% minf(9)
tff(fact_6176_minf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X4),S2))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X4),S2)) ) ) ).

% minf(9)
tff(fact_6177_minf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S2)) ) ) ).

% minf(9)
tff(fact_6178_minf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),S2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),S2)) ) ) ).

% minf(9)
tff(fact_6179_minf_I10_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z2)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S2))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S2)) ) ) ).

% minf(10)
tff(fact_6180_minf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z2)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X4),S2))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X4),S2)) ) ) ).

% minf(10)
tff(fact_6181_minf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S2))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S2)) ) ) ).

% minf(10)
tff(fact_6182_minf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),S2))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),S2)) ) ) ).

% minf(10)
tff(fact_6183_dvd__div__eq__0__iff,axiom,
    ! [B2: complex,A2: complex] :
      ( aa_complex_o(dvd_dvd_complex(B2),A2)
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A2),B2) = zero_zero_complex )
      <=> ( A2 = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6184_dvd__div__eq__0__iff,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) = zero_zero_real )
      <=> ( A2 = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6185_dvd__div__eq__0__iff,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2)
     => ( ( aa_rat_rat(divide_divide_rat(A2),B2) = zero_zero_rat )
      <=> ( A2 = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6186_dvd__div__eq__0__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat )
      <=> ( A2 = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6187_dvd__div__eq__0__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int )
      <=> ( A2 = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_6188_unit__mult__right__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_mult_right_cancel
tff(fact_6189_unit__mult__right__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( ( aa_int_int(times_times_int(B2),A2) = aa_int_int(times_times_int(C2),A2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_mult_right_cancel
tff(fact_6190_unit__mult__left__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_mult_left_cancel
tff(fact_6191_unit__mult__left__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( ( aa_int_int(times_times_int(A2),B2) = aa_int_int(times_times_int(A2),C2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_mult_left_cancel
tff(fact_6192_mult__unit__dvd__iff_H,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C2) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6193_mult__unit__dvd__iff_H,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A2),B2)),C2)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C2) ) ) ).

% mult_unit_dvd_iff'
tff(fact_6194_dvd__mult__unit__iff_H,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6195_dvd__mult__unit__iff_H,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(times_times_int(B2),C2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2) ) ) ).

% dvd_mult_unit_iff'
tff(fact_6196_mult__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2) ) ) ).

% mult_unit_dvd_iff
tff(fact_6197_mult__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A2),B2)),C2)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2) ) ) ).

% mult_unit_dvd_iff
tff(fact_6198_dvd__mult__unit__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2) ) ) ).

% dvd_mult_unit_iff
tff(fact_6199_dvd__mult__unit__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(times_times_int(C2),B2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2) ) ) ).

% dvd_mult_unit_iff
tff(fact_6200_is__unit__mult__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_6201_is__unit__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A2),B2)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_6202_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_6203_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A2),B2)),C2) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_6204_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_6205_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A2),B2)),C2) = aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_6206_unit__div__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),A2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_div_cancel
tff(fact_6207_unit__div__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),A2) )
      <=> ( B2 = C2 ) ) ) ).

% unit_div_cancel
tff(fact_6208_div__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2) ) ) ).

% div_unit_dvd_iff
tff(fact_6209_div__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2) ) ) ).

% div_unit_dvd_iff
tff(fact_6210_dvd__div__unit__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),B2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),C2) ) ) ).

% dvd_div_unit_iff
tff(fact_6211_dvd__div__unit__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),B2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),C2) ) ) ).

% dvd_div_unit_iff
tff(fact_6212_div__mult__div__if__dvd,axiom,
    ! [B2: nat,A2: nat,D2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),C2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),D2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_6213_div__mult__div__if__dvd,axiom,
    ! [B2: int,A2: int,D2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),C2)
       => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),D2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_6214_dvd__mult__imp__div,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),B2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) ) ).

% dvd_mult_imp_div
tff(fact_6215_dvd__mult__imp__div,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A2),C2)),B2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) ) ).

% dvd_mult_imp_div
tff(fact_6216_dvd__div__mult2__eq,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) ) ) ).

% dvd_div_mult2_eq
tff(fact_6217_dvd__div__mult2__eq,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(B2),C2)),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(times_times_int(B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ).

% dvd_div_mult2_eq
tff(fact_6218_div__div__eq__right,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) ) ) ) ).

% div_div_eq_right
tff(fact_6219_div__div__eq__right,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) = aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ) ).

% div_div_eq_right
tff(fact_6220_div__mult__swap,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) ) ) ).

% div_mult_swap
tff(fact_6221_div__mult__swap,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
     => ( aa_int_int(times_times_int(A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A2),B2)),C2) ) ) ).

% div_mult_swap
tff(fact_6222_dvd__div__mult,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),C2) ) ) ).

% dvd_div_mult
tff(fact_6223_dvd__div__mult,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)),A2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(B2),A2)),C2) ) ) ).

% dvd_div_mult
tff(fact_6224_div__power,axiom,
    ! [B2: code_integer,A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
     => ( power_8256067586552552935nteger(divide6298287555418463151nteger(A2,B2),Nb) = divide6298287555418463151nteger(power_8256067586552552935nteger(A2,Nb),power_8256067586552552935nteger(B2,Nb)) ) ) ).

% div_power
tff(fact_6225_div__power,axiom,
    ! [B2: nat,A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb)) ) ) ).

% div_power
tff(fact_6226_div__power,axiom,
    ! [B2: int,A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ( aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),Nb) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(B2),Nb)) ) ) ).

% div_power
tff(fact_6227_dvd__power__le,axiom,
    ! [Xc: nat,Ya: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Ya),M)) ) ) ).

% dvd_power_le
tff(fact_6228_dvd__power__le,axiom,
    ! [Xc: real,Ya: real,Nb: nat,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),M)) ) ) ).

% dvd_power_le
tff(fact_6229_dvd__power__le,axiom,
    ! [Xc: int,Ya: int,Nb: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xc),Nb)),aa_nat_int(power_power_int(Ya),M)) ) ) ).

% dvd_power_le
tff(fact_6230_dvd__power__le,axiom,
    ! [Xc: complex,Ya: complex,Nb: nat,M: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(Xc),Nb)),aa_nat_complex(power_power_complex(Ya),M)) ) ) ).

% dvd_power_le
tff(fact_6231_dvd__power__le,axiom,
    ! [Xc: code_integer,Ya: code_integer,Nb: nat,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xc),Ya)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(Xc,Nb)),power_8256067586552552935nteger(Ya,M)) ) ) ).

% dvd_power_le
tff(fact_6232_power__le__dvd,axiom,
    ! [A2: nat,Nb: nat,B2: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),B2) ) ) ).

% power_le_dvd
tff(fact_6233_power__le__dvd,axiom,
    ! [A2: real,Nb: nat,B2: real,M: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),B2) ) ) ).

% power_le_dvd
tff(fact_6234_power__le__dvd,axiom,
    ! [A2: int,Nb: nat,B2: int,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A2),Nb)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A2),M)),B2) ) ) ).

% power_le_dvd
tff(fact_6235_power__le__dvd,axiom,
    ! [A2: complex,Nb: nat,B2: complex,M: nat] :
      ( aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A2),Nb)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A2),M)),B2) ) ) ).

% power_le_dvd
tff(fact_6236_power__le__dvd,axiom,
    ! [A2: code_integer,Nb: nat,B2: code_integer,M: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(A2,Nb)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(A2,M)),B2) ) ) ).

% power_le_dvd
tff(fact_6237_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)) ) ).

% le_imp_power_dvd
tff(fact_6238_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),M)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ).

% le_imp_power_dvd
tff(fact_6239_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A2),M)),aa_nat_int(power_power_int(A2),Nb)) ) ).

% le_imp_power_dvd
tff(fact_6240_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A2),M)),aa_nat_complex(power_power_complex(A2),Nb)) ) ).

% le_imp_power_dvd
tff(fact_6241_le__imp__power__dvd,axiom,
    ! [M: nat,Nb: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(A2,M)),power_8256067586552552935nteger(A2,Nb)) ) ).

% le_imp_power_dvd
tff(fact_6242_nat__dvd__not__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M) ) ) ).

% nat_dvd_not_less
tff(fact_6243_dvd__minus__self,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ) ).

% dvd_minus_self
tff(fact_6244_less__eq__dvd__minus,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_6245_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_6246_dvd__diffD,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_6247_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_of_int_iff
tff(fact_6248_even__numeral,axiom,
    ! [Nb: num] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_6249_even__numeral,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% even_numeral
tff(fact_6250_unity__coeff__ex,axiom,
    ! [P: fun_complex_o,L: complex] :
      ( ? [X2: complex] : aa_complex_o(P,aa_complex_complex(times_times_complex(L),X2))
    <=> ? [X2: complex] :
          ( aa_complex_o(dvd_dvd_complex(L),aa_complex_complex(plus_plus_complex(X2),zero_zero_complex))
          & aa_complex_o(P,X2) ) ) ).

% unity_coeff_ex
tff(fact_6251_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X2: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X2))
    <=> ? [X2: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X2),zero_zero_real))
          & aa_real_o(P,X2) ) ) ).

% unity_coeff_ex
tff(fact_6252_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X2: rat] : aa_rat_o(P,aa_rat_rat(times_times_rat(L),X2))
    <=> ? [X2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(plus_plus_rat(X2),zero_zero_rat))
          & aa_rat_o(P,X2) ) ) ).

% unity_coeff_ex
tff(fact_6253_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X2: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X2))
    <=> ? [X2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X2),zero_zero_nat))
          & aa_nat_o(P,X2) ) ) ).

% unity_coeff_ex
tff(fact_6254_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X2: int] : aa_int_o(P,aa_int_int(times_times_int(L),X2))
    <=> ? [X2: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(plus_plus_int(X2),zero_zero_int))
          & aa_int_o(P,X2) ) ) ).

% unity_coeff_ex
tff(fact_6255_unit__dvdE,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ~ ( ( A2 != zero_zero_nat )
         => ! [C4: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C4) ) ) ).

% unit_dvdE
tff(fact_6256_unit__dvdE,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ~ ( ( A2 != zero_zero_int )
         => ! [C4: int] : B2 != aa_int_int(times_times_int(A2),C4) ) ) ).

% unit_dvdE
tff(fact_6257_unit__div__eq__0__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat )
      <=> ( A2 = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_6258_unit__div__eq__0__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int )
      <=> ( A2 = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_6259_dvd__div__div__eq__mult,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( C2 != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),D2)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D2),C2) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_6260_dvd__div__div__eq__mult,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( C2 != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),D2)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D2),C2) )
            <=> ( aa_int_int(times_times_int(B2),C2) = aa_int_int(times_times_int(A2),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_6261_dvd__div__iff__mult,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),B2) ) ) ) ).

% dvd_div_iff_mult
tff(fact_6262_dvd__div__iff__mult,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( C2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(times_times_int(A2),C2)),B2) ) ) ) ).

% dvd_div_iff_mult
tff(fact_6263_div__dvd__iff__mult,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_6264_div__dvd__iff__mult,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(times_times_int(C2),B2)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_6265_dvd__div__eq__mult,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A2) = C2 )
        <=> ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_6266_dvd__div__eq__mult,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A2) = C2 )
        <=> ( B2 = aa_int_int(times_times_int(C2),A2) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_6267_inf__period_I3_J,axiom,
    ! [D2: real,D: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),D)
     => ! [X4: real,K5: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),Ta))
        <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D))),Ta)) ) ) ).

% inf_period(3)
tff(fact_6268_inf__period_I3_J,axiom,
    ! [D2: rat,D: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),D)
     => ! [X4: rat,K5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X4),Ta))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K5),D))),Ta)) ) ) ).

% inf_period(3)
tff(fact_6269_inf__period_I3_J,axiom,
    ! [D2: int,D: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X4: int,K5: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),Ta))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D))),Ta)) ) ) ).

% inf_period(3)
tff(fact_6270_inf__period_I4_J,axiom,
    ! [D2: real,D: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),D)
     => ! [X4: real,K5: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),Ta))
        <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K5),D))),Ta)) ) ) ).

% inf_period(4)
tff(fact_6271_inf__period_I4_J,axiom,
    ! [D2: rat,D: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),D)
     => ! [X4: rat,K5: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(X4),Ta))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D2),aa_rat_rat(plus_plus_rat(aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(times_times_rat(K5),D))),Ta)) ) ) ).

% inf_period(4)
tff(fact_6272_inf__period_I4_J,axiom,
    ! [D2: int,D: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X4: int,K5: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),Ta))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),aa_int_int(times_times_int(K5),D))),Ta)) ) ) ).

% inf_period(4)
tff(fact_6273_is__unit__div__mult2__eq,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_6274_is__unit__div__mult2__eq,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(times_times_int(B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_6275_unit__div__mult__swap,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) ) ) ).

% unit_div_mult_swap
tff(fact_6276_unit__div__mult__swap,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),one_one_int)
     => ( aa_int_int(times_times_int(A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A2),B2)),C2) ) ) ).

% unit_div_mult_swap
tff(fact_6277_unit__div__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),B2) ) ) ).

% unit_div_commute
tff(fact_6278_unit__div__commute,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A2),C2)),B2) ) ) ).

% unit_div_commute
tff(fact_6279_div__mult__unit2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2) ) ) ) ).

% div_mult_unit2
tff(fact_6280_div__mult__unit2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(times_times_int(B2),C2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2) ) ) ) ).

% div_mult_unit2
tff(fact_6281_unit__eq__div2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),B2) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2) = C2 ) ) ) ).

% unit_eq_div2
tff(fact_6282_unit__eq__div2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( ( A2 = aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),B2) )
      <=> ( aa_int_int(times_times_int(A2),B2) = C2 ) ) ) ).

% unit_eq_div2
tff(fact_6283_unit__eq__div1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = C2 )
      <=> ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2) ) ) ) ).

% unit_eq_div1
tff(fact_6284_unit__eq__div1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = C2 )
      <=> ( A2 = aa_int_int(times_times_int(C2),B2) ) ) ) ).

% unit_eq_div1
tff(fact_6285_is__unit__power__iff,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6286_is__unit__power__iff,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A2),Nb)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6287_is__unit__power__iff,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(A2,Nb)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),one_one_Code_integer)
        | ( Nb = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_6288_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% dvd_imp_le
tff(fact_6289_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_6290_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_6291_real__of__nat__div,axiom,
    ! [D2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),D2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,D2)) ) ) ).

% real_of_nat_div
tff(fact_6292_even__zero,axiom,
    aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_zero_nat) ).

% even_zero
tff(fact_6293_even__zero,axiom,
    aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),zero_zero_int) ).

% even_zero
tff(fact_6294_odd__even__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)) ) ) ).

% odd_even_add
tff(fact_6295_odd__even__add,axiom,
    ! [A2: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(A2),B2)) ) ) ).

% odd_even_add
tff(fact_6296_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),one_one_nat) ).

% odd_one
tff(fact_6297_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),one_one_int) ).

% odd_one
tff(fact_6298_evenE,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ~ ! [B3: nat] : A2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_6299_evenE,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ~ ! [B3: int] : A2 != aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3) ) ).

% evenE
tff(fact_6300_bit__eq__rec,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2) )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_6301_bit__eq__rec,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
    <=> ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_6302_is__unit__div__mult__cancel__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_6303_is__unit__div__mult__cancel__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(times_times_int(B2),A2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_6304_is__unit__div__mult__cancel__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_6305_is__unit__div__mult__cancel__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(times_times_int(A2),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_6306_is__unitE,axiom,
    ! [A2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),one_one_nat)
     => ~ ( ( A2 != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B3),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A2) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A2 )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B3) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C2),A2) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_6307_is__unitE,axiom,
    ! [A2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),one_one_int)
     => ~ ( ( A2 != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B3),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A2) = B3 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B3) = A2 )
                   => ( ( aa_int_int(times_times_int(A2),B3) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C2),A2) != aa_int_int(times_times_int(C2),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_6308_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6309_odd__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% odd_numeral
tff(fact_6310_dvd__power__iff,axiom,
    ! [Xc: nat,M: nat,Nb: nat] :
      ( ( Xc != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xc),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_6311_dvd__power__iff,axiom,
    ! [Xc: int,M: nat,Nb: nat] :
      ( ( Xc != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xc),M)),aa_nat_int(power_power_int(Xc),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xc),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_6312_dvd__power__iff,axiom,
    ! [Xc: code_integer,M: nat,Nb: nat] :
      ( ( Xc != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(Xc,M)),power_8256067586552552935nteger(Xc,Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xc),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_6313_dvd__power,axiom,
    ! [Nb: nat,Xc: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xc = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xc),aa_nat_rat(power_power_rat(Xc),Nb)) ) ).

% dvd_power
tff(fact_6314_dvd__power,axiom,
    ! [Nb: nat,Xc: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xc = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xc),Nb)) ) ).

% dvd_power
tff(fact_6315_dvd__power,axiom,
    ! [Nb: nat,Xc: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xc = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xc),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb)) ) ).

% dvd_power
tff(fact_6316_dvd__power,axiom,
    ! [Nb: nat,Xc: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xc = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xc),aa_nat_int(power_power_int(Xc),Nb)) ) ).

% dvd_power
tff(fact_6317_dvd__power,axiom,
    ! [Nb: nat,Xc: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xc = one_one_complex ) )
     => aa_complex_o(dvd_dvd_complex(Xc),aa_nat_complex(power_power_complex(Xc),Nb)) ) ).

% dvd_power
tff(fact_6318_dvd__power,axiom,
    ! [Nb: nat,Xc: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xc = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xc),power_8256067586552552935nteger(Xc,Nb)) ) ).

% dvd_power
tff(fact_6319_dvd__mult__cancel2,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),M)),M)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_6320_dvd__mult__cancel1,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)),M)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_6321_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_6322_dvd__minus__add,axiom,
    ! [Q3: nat,Nb: nat,R3: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R3),M))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q3))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R3),M)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_6323_even__two__times__div__two,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = A2 ) ) ).

% even_two_times_div_two
tff(fact_6324_even__two__times__div__two,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A2 ) ) ).

% even_two_times_div_two
tff(fact_6325_power__mono__odd,axiom,
    ! [Nb: nat,A2: real,B2: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb)) ) ) ).

% power_mono_odd
tff(fact_6326_power__mono__odd,axiom,
    ! [Nb: nat,A2: code_integer,B2: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,Nb)),power_8256067586552552935nteger(B2,Nb)) ) ) ).

% power_mono_odd
tff(fact_6327_power__mono__odd,axiom,
    ! [Nb: nat,A2: rat,B2: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),Nb)),aa_nat_rat(power_power_rat(B2),Nb)) ) ) ).

% power_mono_odd
tff(fact_6328_power__mono__odd,axiom,
    ! [Nb: nat,A2: int,B2: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(B2),Nb)) ) ) ).

% power_mono_odd
tff(fact_6329_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% odd_pos
tff(fact_6330_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_6331_even__set__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(M),A2))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_6332_even__set__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),A2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_6333_oddE,axiom,
    ! [A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ~ ! [B3: nat] : A2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_6334_oddE,axiom,
    ! [A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ~ ! [B3: int] : A2 != aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B3)),one_one_int) ) ).

% oddE
tff(fact_6335_zero__le__even__power,axiom,
    ! [Nb: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ).

% zero_le_even_power
tff(fact_6336_zero__le__even__power,axiom,
    ! [Nb: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,Nb)) ) ).

% zero_le_even_power
tff(fact_6337_zero__le__even__power,axiom,
    ! [Nb: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),Nb)) ) ).

% zero_le_even_power
tff(fact_6338_zero__le__even__power,axiom,
    ! [Nb: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),Nb)) ) ).

% zero_le_even_power
tff(fact_6339_zero__le__odd__power,axiom,
    ! [Nb: nat,A2: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ) ).

% zero_le_odd_power
tff(fact_6340_zero__le__odd__power,axiom,
    ! [Nb: nat,A2: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,Nb))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ) ).

% zero_le_odd_power
tff(fact_6341_zero__le__odd__power,axiom,
    ! [Nb: nat,A2: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),Nb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ) ).

% zero_le_odd_power
tff(fact_6342_zero__le__odd__power,axiom,
    ! [Nb: nat,A2: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),Nb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ) ).

% zero_le_odd_power
tff(fact_6343_zero__le__power__eq,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ) ) ).

% zero_le_power_eq
tff(fact_6344_zero__le__power__eq,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ) ) ).

% zero_le_power_eq
tff(fact_6345_zero__le__power__eq,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ) ) ).

% zero_le_power_eq
tff(fact_6346_zero__le__power__eq,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(A2),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ) ) ).

% zero_le_power_eq
tff(fact_6347_zero__less__power__eq,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(A2,Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A2 != zero_z3403309356797280102nteger ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2) ) ) ) ).

% zero_less_power_eq
tff(fact_6348_zero__less__power__eq,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A2 != zero_zero_real ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ) ) ).

% zero_less_power_eq
tff(fact_6349_zero__less__power__eq,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(A2),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A2 != zero_zero_rat ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ) ) ).

% zero_less_power_eq
tff(fact_6350_zero__less__power__eq,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(A2),Nb))
    <=> ( ( Nb = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & ( A2 != zero_zero_int ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ) ) ).

% zero_less_power_eq
tff(fact_6351_even__mask__div__iff_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M),one_one_Code_integer),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% even_mask_div_iff'
tff(fact_6352_even__mask__div__iff_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% even_mask_div_iff'
tff(fact_6353_even__mask__div__iff_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% even_mask_div_iff'
tff(fact_6354_power__le__zero__eq,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),zero_zero_real)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A2 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6355_power__le__zero__eq,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,Nb)),zero_z3403309356797280102nteger)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A2 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6356_power__le__zero__eq,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),Nb)),zero_zero_rat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A2 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6357_power__le__zero__eq,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),Nb)),zero_zero_int)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        & ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) )
          | ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
            & ( A2 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_6358_even__mask__div__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M),one_one_Code_integer),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)))
    <=> ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% even_mask_div_iff
tff(fact_6359_even__mask__div__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% even_mask_div_iff
tff(fact_6360_even__mask__div__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ) ).

% even_mask_div_iff
tff(fact_6361_divmod__divmod__step,axiom,
    ! [M: num,Nb: num] :
      unique5055182867167087721od_nat(M,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,M)),unique5026877609467782581ep_nat(Nb,unique5055182867167087721od_nat(M,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6362_divmod__divmod__step,axiom,
    ! [M: num,Nb: num] :
      unique5052692396658037445od_int(M,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,M)),unique5024387138958732305ep_int(Nb,unique5052692396658037445od_int(M,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6363_divmod__divmod__step,axiom,
    ! [M: num,Nb: num] :
      unique3479559517661332726nteger(M,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,M),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(M)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(M,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_6364_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_6365_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_6366_even__mult__exp__div__exp__iff,axiom,
    ! [A2: code_integer,M: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(times_3573771949741848930nteger(A2,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A2,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6367_even__mult__exp__div__exp__iff,axiom,
    ! [A2: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6368_even__mult__exp__div__exp__iff,axiom,
    ! [A2: int,M: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_6369_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xc) = Ya )
     => ( ( ( Xc = zero_zero_nat )
         => ( Ya != aa_nat_nat(suc,zero_zero_nat) ) )
       => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
           => ( Ya != aa_nat_nat(suc,zero_zero_nat) ) )
         => ~ ! [N: nat] :
                ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Ya != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_6370_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_6371_VEBT__internal_OTb_H_Oelims,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,Xc) = Ya )
     => ( ( ( Xc = zero_zero_nat )
         => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
           => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [N: nat] :
                ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Ya != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_6372_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% VEBT_internal.Tb.simps(3)
tff(fact_6373_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_6374_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Vaa: nat] :
      aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_6375_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Vaa: nat] :
      aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
          $let(
            half: nat,
            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_6376_VEBT__internal_OTb_Oelims,axiom,
    ! [Xc: nat,Ya: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,Xc) = Ya )
     => ( ( ( Xc = zero_zero_nat )
         => ( Ya != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
       => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
           => ( Ya != aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) ) )
         => ~ ! [N: nat] :
                ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
               => ( Ya != $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_6377_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xc) = Ya )
     => ( ( ( Xc = zero_zero_nat )
         => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
       => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
           => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Ya != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_6378_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xc) = Ya )
     => ( ( ( Xc = zero_zero_nat )
         => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
       => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
           => ( Ya != aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ) )
         => ~ ! [Va2: nat] :
                ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Ya != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                        $let(
                          half: nat,
                          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_6379_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2) ) ) ).

% pow_divides_pow_iff
tff(fact_6380_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A2: int,B2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(B2),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2) ) ) ).

% pow_divides_pow_iff
tff(fact_6381_artanh__def,axiom,
    ! [Xc: real] : aa_real_real(artanh_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xc)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_6382_div2__even__ext__nat,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xc)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Ya) )
       => ( Xc = Ya ) ) ) ).

% div2_even_ext_nat
tff(fact_6383_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [D5: nat,X3: nat,Y: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D5),A2)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D5),B2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y)),D5) ) ) ) ).

% bezout_add_strong_nat
tff(fact_6384_highi__def,axiom,
    ! [Xc: nat,Nb: nat] : vEBT_VEBT_highi(Xc,Nb) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% highi_def
tff(fact_6385_nth__rule,axiom,
    ! [I: nat,Xs: list_o,A2: array_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_o(Xs))
     => hoare_hoare_triple_o(snga_assn_o(A2,Xs),array_nth_o(A2,I),aa_arr2544151649598861476o_assn(aa_lis966790859614740569o_assn(aTP_Lamm_ee(I),Xs),A2)) ) ).

% nth_rule
tff(fact_6386_nth__rule,axiom,
    ! [I: nat,Xs: list_VEBT_VEBTi,A2: array_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s7982070591426661849_VEBTi(Xs))
     => hoare_1429296392585015714_VEBTi(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(A2),Xs),array_nth_VEBT_VEBTi(A2,I),aa_arr8081119733880211216i_assn(aa_lis5880058344902901532i_assn(aTP_Lamm_ef(I),Xs),A2)) ) ).

% nth_rule
tff(fact_6387_nth__rule,axiom,
    ! [I: nat,Xs: list_option_nat,A2: array_option_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6086282163384603972on_nat(Xs))
     => hoare_7629718768684598413on_nat(snga_assn_option_nat(A2,Xs),array_nth_option_nat(A2,I),aa_arr7483714166967448038t_assn(aa_lis7582702936096254407t_assn(aTP_Lamm_eg(I),Xs),A2)) ) ).

% nth_rule
tff(fact_6388_nth__rule,axiom,
    ! [I: nat,Xs: list_nat,A2: array_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Xs))
     => hoare_3067605981109127869le_nat(snga_assn_nat(A2,Xs),array_nth_nat(A2,I),aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_eh(I),Xs),A2)) ) ).

% nth_rule
tff(fact_6389_highsimp,axiom,
    ! [Xc: nat,Nb: nat] : heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xc),Nb)) = vEBT_VEBT_highi(Xc,Nb) ).

% highsimp
tff(fact_6390_lowsimp,axiom,
    ! [Xc: nat,Nb: nat] : heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),Nb)) = vEBT_VEBT_lowi(Xc,Nb) ).

% lowsimp
tff(fact_6391_ln__inj__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => ( ( aa_real_real(ln_ln_real,Xc) = aa_real_real(ln_ln_real,Ya) )
        <=> ( Xc = Ya ) ) ) ) ).

% ln_inj_iff
tff(fact_6392_ln__less__cancel__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xc)),aa_real_real(ln_ln_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ) ) ).

% ln_less_cancel_iff
tff(fact_6393_ln__le__cancel__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xc)),aa_real_real(ln_ln_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ) ) ).

% ln_le_cancel_iff
tff(fact_6394_ln__eq__zero__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( ( aa_real_real(ln_ln_real,Xc) = zero_zero_real )
      <=> ( Xc = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_6395_ln__gt__zero__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xc))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc) ) ) ).

% ln_gt_zero_iff
tff(fact_6396_ln__less__zero__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xc)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_6397_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_6398_ln__ge__zero__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xc))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc) ) ) ).

% ln_ge_zero_iff
tff(fact_6399_ln__le__zero__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xc)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_6400_zdvd__zdiffD,axiom,
    ! [K: int,M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(minus_minus_int(M),Nb))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M) ) ) ).

% zdvd_zdiffD
tff(fact_6401_ln__less__self,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xc)),Xc) ) ).

% ln_less_self
tff(fact_6402_log__def,axiom,
    ! [A2: real,Xc: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xc)),aa_real_real(ln_ln_real,A2)) ).

% log_def
tff(fact_6403_zdvd__not__zless,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,M),Nb)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),M) ) ) ).

% zdvd_not_zless
tff(fact_6404_ln__bound,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xc)),Xc) ) ).

% ln_bound
tff(fact_6405_ln__gt__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xc)) ) ).

% ln_gt_zero
tff(fact_6406_ln__less__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xc)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_6407_ln__gt__zero__imp__gt__one,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xc))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_6408_ln__ge__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xc)) ) ).

% ln_ge_zero
tff(fact_6409_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_6410_real__of__int__div,axiom,
    ! [D2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),Nb)
     => ( aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),D2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,D2)) ) ) ).

% real_of_int_div
tff(fact_6411_prod__decode__aux_Ocases,axiom,
    ! [Xc: product_prod_nat_nat] :
      ~ ! [K2: nat,M4: nat] : Xc != aa_nat7603221163087113075at_nat(product_Pair_nat_nat(K2),M4) ).

% prod_decode_aux.cases
tff(fact_6412_ln__ge__zero__imp__ge__one,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xc))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_6413_ln__add__one__self__le__self,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc))),Xc) ) ).

% ln_add_one_self_le_self
tff(fact_6414_ln__mult,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xc)),aa_real_real(ln_ln_real,Ya)) ) ) ) ).

% ln_mult
tff(fact_6415_ln__eq__minus__one,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( ( aa_real_real(ln_ln_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),one_one_real) )
       => ( Xc = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_6416_ln__div,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xc)),aa_real_real(ln_ln_real,Ya)) ) ) ) ).

% ln_div
tff(fact_6417_int__div__sub__1,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),M)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(Nb),one_one_int)),M) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),Nb),aa_int_int(minus_minus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),M)),one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),M)) ) ) ).

% int_div_sub_1
tff(fact_6418_bset_I9_J,axiom,
    ! [D2: int,D: int,B: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B)
                 => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),D)),Ta)) ) ) ) ).

% bset(9)
tff(fact_6419_bset_I10_J,axiom,
    ! [D2: int,D: int,B: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),B)
                 => ( X4 != aa_int_int(plus_plus_int(Xb2),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(X4),D)),Ta)) ) ) ) ).

% bset(10)
tff(fact_6420_aset_I9_J,axiom,
    ! [D2: int,D: int,A: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A)
                 => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X4),D)),Ta)) ) ) ) ).

% aset(9)
tff(fact_6421_aset_I10_J,axiom,
    ! [D2: int,D: int,A: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( aa_set_int_o(member_int(Xa2),set_or1266510415728281911st_int(one_one_int,D))
             => ! [Xb2: int] :
                  ( aa_set_int_o(member_int(Xb2),A)
                 => ( X4 != aa_int_int(minus_minus_int(Xb2),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(X4),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(X4),D)),Ta)) ) ) ) ).

% aset(10)
tff(fact_6422_ln__le__minus__one,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_6423_ln__diff__le,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xc)),aa_real_real(ln_ln_real,Ya))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Ya)),Ya)) ) ) ).

% ln_diff_le
tff(fact_6424_ln__realpow,axiom,
    ! [Xc: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,Xc)) ) ) ).

% ln_realpow
tff(fact_6425_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(minus_minus_int(K),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),L)) ) ).

% even_diff_iff
tff(fact_6426_return__sp__rule,axiom,
    ! [P: assn,Xc: $o] : hoare_hoare_triple_o(P,heap_Time_return_o((Xc)),aa_o_fun_o_assn(aTP_Lamm_ei(P),(Xc))) ).

% return_sp_rule
tff(fact_6427_return__sp__rule,axiom,
    ! [P: assn,Xc: vEBT_VEBTi] : hoare_1429296392585015714_VEBTi(P,heap_T3630416162098727440_VEBTi(Xc),aa_VEB1545114916007340237i_assn(aTP_Lamm_ej(P),Xc)) ).

% return_sp_rule
tff(fact_6428_return__sp__rule,axiom,
    ! [P: assn,Xc: option_nat] : hoare_7629718768684598413on_nat(P,heap_T3487192422709364219on_nat(Xc),aa_opt3861602675778688675t_assn(aTP_Lamm_ek(P),Xc)) ).

% return_sp_rule
tff(fact_6429_return__sp__rule,axiom,
    ! [P: assn,Xc: nat] : hoare_3067605981109127869le_nat(P,heap_Time_return_nat(Xc),aa_nat_fun_nat_assn(aTP_Lamm_el(P),Xc)) ).

% return_sp_rule
tff(fact_6430_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
         => ( ( B2 != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B2)),aa_real_real(ln_ln_real,A2))),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_6431_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_int_nat(nat2,Z)),M)
    <=> $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z),aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Z),aa_nat_int(semiri1314217659103216013at_int,M)),M = zero_zero_nat) ) ).

% nat_dvd_iff
tff(fact_6432_list__decode_Ocases,axiom,
    ! [Xc: nat] :
      ( ( Xc != zero_zero_nat )
     => ~ ! [N: nat] : Xc != aa_nat_nat(suc,N) ) ).

% list_decode.cases
tff(fact_6433_dvd__productE,axiom,
    ! [P3: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,P3),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2))
     => ~ ! [X3: nat,Y: nat] :
            ( ( P3 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X3),Y) )
           => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X3),A2)
             => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Y),B2) ) ) ) ).

% dvd_productE
tff(fact_6434_dvd__productE,axiom,
    ! [P3: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P3),aa_int_int(times_times_int(A2),B2))
     => ~ ! [X3: int,Y: int] :
            ( ( P3 = aa_int_int(times_times_int(X3),Y) )
           => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X3),A2)
             => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y),B2) ) ) ) ).

% dvd_productE
tff(fact_6435_division__decomp,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))
     => ? [B7: nat,C6: nat] :
          ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B7),C6) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B7),B2)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C6),C2) ) ) ).

% division_decomp
tff(fact_6436_division__decomp,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),aa_int_int(times_times_int(B2),C2))
     => ? [B7: int,C6: int] :
          ( ( A2 = aa_int_int(times_times_int(B7),C6) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B7),B2)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C6),C2) ) ) ).

% division_decomp
tff(fact_6437_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B3)
        <=> aa_nat_o(aa_nat_fun_nat_o(P,B3),A3) )
     => ( ! [A3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,A3),zero_zero_nat)
       => ( ! [A3: nat,B3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(P,A3),B3)
             => aa_nat_o(aa_nat_fun_nat_o(P,A3),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),B3)) )
         => aa_nat_o(aa_nat_fun_nat_o(P,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_6438_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),zero_zero_nat) ).

% gcd_nat.extremum
tff(fact_6439_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A2)
        & ( zero_zero_nat != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_6440_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A2)
    <=> ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
tff(fact_6441_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),zero_zero_nat)
        & ( A2 != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_6442_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,zero_zero_nat),A2)
     => ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_6443_even__nat__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_int_nat(nat2,K))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ) ).

% even_nat_iff
tff(fact_6444_ln__one__plus__pos__lower__bound,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_6445_dvd__pos__nat,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M) ) ) ).

% dvd_pos_nat
tff(fact_6446_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D5: nat,X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D5),A2)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D5),B2)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y)),D5) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y)),D5) ) ) ) ).

% bezout_add_nat
tff(fact_6447_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat,Xc: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B2)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Xc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Ya)),D2) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Xc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Ya)),D2) ) )
         => ? [X3: nat,Y: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A2)
              & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))
              & ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),Y)),D2) )
                | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),X3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_6448_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D5: nat,X3: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D5),A2)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D5),B2)
      & ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y)) = D5 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),Y)) = D5 ) ) ) ).

% bezout1_nat
tff(fact_6449_ln__2__less__1,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),one_one_real) ).

% ln_2_less_1
tff(fact_6450_lowi__def,axiom,
    ! [Xc: nat,Nb: nat] : vEBT_VEBT_lowi(Xc,Nb) = heap_Time_return_nat(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% lowi_def
tff(fact_6451_unset__bit__0,axiom,
    ! [A2: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A2) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_6452_unset__bit__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_6453_fi__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_o,Q: fun_o_assn,Ps: assn,F: assn] :
      ( hoare_hoare_triple_o(P,C2,Q)
     => ( entails(Ps,aa_assn_assn(times_times_assn(P),F))
       => hoare_hoare_triple_o(Ps,C2,aa_assn_fun_o_assn(aTP_Lamm_da(Q),F)) ) ) ).

% fi_rule
tff(fact_6454_fi__rule,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,Ps: assn,F: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C2,Q)
     => ( entails(Ps,aa_assn_assn(times_times_assn(P),F))
       => hoare_1429296392585015714_VEBTi(Ps,C2,aa_ass4389913640333755968i_assn(aTP_Lamm_db(Q),F)) ) ) ).

% fi_rule
tff(fact_6455_fi__rule,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,Ps: assn,F: assn] :
      ( hoare_7629718768684598413on_nat(P,C2,Q)
     => ( entails(Ps,aa_assn_assn(times_times_assn(P),F))
       => hoare_7629718768684598413on_nat(Ps,C2,aa_ass4112278883933122603t_assn(aTP_Lamm_dc(Q),F)) ) ) ).

% fi_rule
tff(fact_6456_fi__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,Ps: assn,F: assn] :
      ( hoare_3067605981109127869le_nat(P,C2,Q)
     => ( entails(Ps,aa_assn_assn(times_times_assn(P),F))
       => hoare_3067605981109127869le_nat(Ps,C2,aa_assn_fun_nat_assn(aTP_Lamm_dd(Q),F)) ) ) ).

% fi_rule
tff(fact_6457_low__def,axiom,
    ! [Xc: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xc),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% low_def
tff(fact_6458_return__cons__rule,axiom,
    ! [P: assn,Q: fun_o_assn,Xc: $o] :
      ( entails(P,aa_o_assn(Q,(Xc)))
     => hoare_hoare_triple_o(P,heap_Time_return_o((Xc)),Q) ) ).

% return_cons_rule
tff(fact_6459_return__cons__rule,axiom,
    ! [P: assn,Q: fun_VEBT_VEBTi_assn,Xc: vEBT_VEBTi] :
      ( entails(P,aa_VEBT_VEBTi_assn(Q,Xc))
     => hoare_1429296392585015714_VEBTi(P,heap_T3630416162098727440_VEBTi(Xc),Q) ) ).

% return_cons_rule
tff(fact_6460_return__cons__rule,axiom,
    ! [P: assn,Q: fun_option_nat_assn,Xc: option_nat] :
      ( entails(P,aa_option_nat_assn(Q,Xc))
     => hoare_7629718768684598413on_nat(P,heap_T3487192422709364219on_nat(Xc),Q) ) ).

% return_cons_rule
tff(fact_6461_return__cons__rule,axiom,
    ! [P: assn,Q: fun_nat_assn,Xc: nat] :
      ( entails(P,aa_nat_assn(Q,Xc))
     => hoare_3067605981109127869le_nat(P,heap_Time_return_nat(Xc),Q) ) ).

% return_cons_rule
tff(fact_6462_mod__mod__trivial,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mod_trivial
tff(fact_6463_mod__mod__trivial,axiom,
    ! [A2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A2),B2)),B2) = aa_int_int(modulo_modulo_int(A2),B2) ).

% mod_mod_trivial
tff(fact_6464_mod__mod__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(A2,B2),B2) = modulo364778990260209775nteger(A2,B2) ).

% mod_mod_trivial
tff(fact_6465_bits__mod__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A2) = zero_zero_nat ).

% bits_mod_0
tff(fact_6466_bits__mod__0,axiom,
    ! [A2: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A2) = zero_zero_int ).

% bits_mod_0
tff(fact_6467_bits__mod__0,axiom,
    ! [A2: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A2) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_6468_mod__self,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),A2) = zero_zero_nat ).

% mod_self
tff(fact_6469_mod__self,axiom,
    ! [A2: int] : aa_int_int(modulo_modulo_int(A2),A2) = zero_zero_int ).

% mod_self
tff(fact_6470_mod__self,axiom,
    ! [A2: code_integer] : modulo364778990260209775nteger(A2,A2) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_6471_mod__by__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),zero_zero_nat) = A2 ).

% mod_by_0
tff(fact_6472_mod__by__0,axiom,
    ! [A2: int] : aa_int_int(modulo_modulo_int(A2),zero_zero_int) = A2 ).

% mod_by_0
tff(fact_6473_mod__by__0,axiom,
    ! [A2: code_integer] : modulo364778990260209775nteger(A2,zero_z3403309356797280102nteger) = A2 ).

% mod_by_0
tff(fact_6474_mod__0,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,zero_zero_nat),A2) = zero_zero_nat ).

% mod_0
tff(fact_6475_mod__0,axiom,
    ! [A2: int] : aa_int_int(modulo_modulo_int(zero_zero_int),A2) = zero_zero_int ).

% mod_0
tff(fact_6476_mod__0,axiom,
    ! [A2: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A2) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_6477_mod__add__self2,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_add_self2
tff(fact_6478_mod__add__self2,axiom,
    ! [A2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),B2)),B2) = aa_int_int(modulo_modulo_int(A2),B2) ).

% mod_add_self2
tff(fact_6479_mod__add__self2,axiom,
    ! [A2: code_integer,B2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A2,B2),B2) = modulo364778990260209775nteger(A2,B2) ).

% mod_add_self2
tff(fact_6480_mod__add__self1,axiom,
    ! [B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_add_self1
tff(fact_6481_mod__add__self1,axiom,
    ! [B2: int,A2: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(B2),A2)),B2) = aa_int_int(modulo_modulo_int(A2),B2) ).

% mod_add_self1
tff(fact_6482_mod__add__self1,axiom,
    ! [B2: code_integer,A2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B2,A2),B2) = modulo364778990260209775nteger(A2,B2) ).

% mod_add_self1
tff(fact_6483_minus__mod__self2,axiom,
    ! [A2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A2),B2)),B2) = aa_int_int(modulo_modulo_int(A2),B2) ).

% minus_mod_self2
tff(fact_6484_minus__mod__self2,axiom,
    ! [A2: code_integer,B2: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A2,B2),B2) = modulo364778990260209775nteger(A2,B2) ).

% minus_mod_self2
tff(fact_6485_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se4203085406695923979it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_6486_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se4203085406695923979it_int(Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% unset_bit_negative_int_iff
tff(fact_6487_nat__mod__eq_H,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb) = A2 ) ) ).

% nat_mod_eq'
tff(fact_6488_mod__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = M ) ) ).

% mod_less
tff(fact_6489_bits__mod__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_6490_bits__mod__by__1,axiom,
    ! [A2: int] : aa_int_int(modulo_modulo_int(A2),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_6491_bits__mod__by__1,axiom,
    ! [A2: code_integer] : modulo364778990260209775nteger(A2,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_6492_mod__by__1,axiom,
    ! [A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_6493_mod__by__1,axiom,
    ! [A2: int] : aa_int_int(modulo_modulo_int(A2),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_6494_mod__by__1,axiom,
    ! [A2: code_integer] : modulo364778990260209775nteger(A2,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_6495_mod__mult__self2__is__0,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),B2) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_6496_mod__mult__self2__is__0,axiom,
    ! [A2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A2),B2)),B2) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_6497_mod__mult__self2__is__0,axiom,
    ! [A2: code_integer,B2: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(A2,B2),B2) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_6498_mod__mult__self1__is__0,axiom,
    ! [B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A2)),B2) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_6499_mod__mult__self1__is__0,axiom,
    ! [B2: int,A2: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(B2),A2)),B2) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_6500_mod__mult__self1__is__0,axiom,
    ! [B2: code_integer,A2: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(B2,A2),B2) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_6501_bits__mod__div__trivial,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_6502_bits__mod__div__trivial,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(modulo_modulo_int(A2),B2)),B2) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_6503_bits__mod__div__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A2,B2),B2) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_6504_mod__div__trivial,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2) = zero_zero_nat ).

% mod_div_trivial
tff(fact_6505_mod__div__trivial,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(modulo_modulo_int(A2),B2)),B2) = zero_zero_int ).

% mod_div_trivial
tff(fact_6506_mod__div__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A2,B2),B2) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_6507_mod__mult__self4,axiom,
    ! [B2: nat,C2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self4
tff(fact_6508_mod__mult__self4,axiom,
    ! [B2: int,C2: int,A2: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),C2)),A2)),B2) = aa_int_int(modulo_modulo_int(A2),B2) ).

% mod_mult_self4
tff(fact_6509_mod__mult__self4,axiom,
    ! [B2: code_integer,C2: code_integer,A2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(B2,C2),A2),B2) = modulo364778990260209775nteger(A2,B2) ).

% mod_mult_self4
tff(fact_6510_mod__mult__self3,axiom,
    ! [C2: nat,B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)),A2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self3
tff(fact_6511_mod__mult__self3,axiom,
    ! [C2: int,B2: int,A2: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(C2),B2)),A2)),B2) = aa_int_int(modulo_modulo_int(A2),B2) ).

% mod_mult_self3
tff(fact_6512_mod__mult__self3,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(C2,B2),A2),B2) = modulo364778990260209775nteger(A2,B2) ).

% mod_mult_self3
tff(fact_6513_mod__mult__self2,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self2
tff(fact_6514_mod__mult__self2,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),aa_int_int(times_times_int(B2),C2))),B2) = aa_int_int(modulo_modulo_int(A2),B2) ).

% mod_mult_self2
tff(fact_6515_mod__mult__self2,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A2,times_3573771949741848930nteger(B2,C2)),B2) = modulo364778990260209775nteger(A2,B2) ).

% mod_mult_self2
tff(fact_6516_mod__mult__self1,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% mod_mult_self1
tff(fact_6517_mod__mult__self1,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),aa_int_int(times_times_int(C2),B2))),B2) = aa_int_int(modulo_modulo_int(A2),B2) ).

% mod_mult_self1
tff(fact_6518_mod__mult__self1,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A2,times_3573771949741848930nteger(C2,B2)),B2) = modulo364778990260209775nteger(A2,B2) ).

% mod_mult_self1
tff(fact_6519_dvd__imp__mod__0,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),A2) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_6520_dvd__imp__mod__0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
     => ( aa_int_int(modulo_modulo_int(B2),A2) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_6521_dvd__imp__mod__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
     => ( modulo364778990260209775nteger(B2,A2) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_6522_mod__by__Suc__0,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_6523_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)),M))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% Suc_mod_mult_self4
tff(fact_6524_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),M))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% Suc_mod_mult_self3
tff(fact_6525_Suc__mod__mult__self2,axiom,
    ! [M: nat,Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% Suc_mod_mult_self2
tff(fact_6526_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% Suc_mod_mult_self1
tff(fact_6527_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_6528_bits__one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_6529_bits__one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_6530_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_6531_one__mod__two__eq__one,axiom,
    aa_int_int(modulo_modulo_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_6532_one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_6533_even__mod__2__iff,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2) ) ).

% even_mod_2_iff
tff(fact_6534_even__mod__2__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2) ) ).

% even_mod_2_iff
tff(fact_6535_even__mod__2__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2) ) ).

% even_mod_2_iff
tff(fact_6536_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_6537_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) != one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),Nb))),aa_num_nat(numeral_numeral_nat,K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_6538_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_6539_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: int] :
      ( ( aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_6540_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_6541_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_6542_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: int] :
      ( ( aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_6543_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_6544_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != aa_nat_nat(suc,zero_zero_nat) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_6545_add__self__mod__2,axiom,
    ! [M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),M)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_6546_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_6547_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_6548_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_6549_even__succ__mod__exp,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6550_even__succ__mod__exp,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),A2)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(modulo_modulo_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6551_even__succ__mod__exp,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A2),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A2,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_6552_of__nat__mod,axiom,
    ! [M: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = aa_int_int(modulo_modulo_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mod
tff(fact_6553_of__nat__mod,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mod
tff(fact_6554_of__nat__mod,axiom,
    ! [M: nat,Nb: nat] : aa_nat_Code_integer(semiri4939895301339042750nteger,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = modulo364778990260209775nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) ).

% of_nat_mod
tff(fact_6555_mod__add__right__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) ).

% mod_add_right_eq
tff(fact_6556_mod__add__right__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),aa_int_int(modulo_modulo_int(B2),C2))),C2) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),B2)),C2) ).

% mod_add_right_eq
tff(fact_6557_mod__add__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A2,modulo364778990260209775nteger(B2,C2)),C2) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A2,B2),C2) ).

% mod_add_right_eq
tff(fact_6558_mod__add__left__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2)),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) ).

% mod_add_left_eq
tff(fact_6559_mod__add__left__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A2),C2)),B2)),C2) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),B2)),C2) ).

% mod_add_left_eq
tff(fact_6560_mod__add__left__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A2,C2),B2),C2) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A2,B2),C2) ).

% mod_add_left_eq
tff(fact_6561_mod__add__cong,axiom,
    ! [A2: nat,C2: nat,A4: nat,B2: nat,B4: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),C2) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),C2) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A4),B4)),C2) ) ) ) ).

% mod_add_cong
tff(fact_6562_mod__add__cong,axiom,
    ! [A2: int,C2: int,A4: int,B2: int,B4: int] :
      ( ( aa_int_int(modulo_modulo_int(A2),C2) = aa_int_int(modulo_modulo_int(A4),C2) )
     => ( ( aa_int_int(modulo_modulo_int(B2),C2) = aa_int_int(modulo_modulo_int(B4),C2) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),B2)),C2) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A4),B4)),C2) ) ) ) ).

% mod_add_cong
tff(fact_6563_mod__add__cong,axiom,
    ! [A2: code_integer,C2: code_integer,A4: code_integer,B2: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A2,C2) = modulo364778990260209775nteger(A4,C2) )
     => ( ( modulo364778990260209775nteger(B2,C2) = modulo364778990260209775nteger(B4,C2) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A2,B2),C2) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A4,B4),C2) ) ) ) ).

% mod_add_cong
tff(fact_6564_mod__add__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) ).

% mod_add_eq
tff(fact_6565_mod__add__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A2),C2)),aa_int_int(modulo_modulo_int(B2),C2))),C2) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),B2)),C2) ).

% mod_add_eq
tff(fact_6566_mod__add__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A2,C2),modulo364778990260209775nteger(B2,C2)),C2) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A2,B2),C2) ).

% mod_add_eq
tff(fact_6567_mod__mult__right__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) ).

% mod_mult_right_eq
tff(fact_6568_mod__mult__right__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A2),aa_int_int(modulo_modulo_int(B2),C2))),C2) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A2),B2)),C2) ).

% mod_mult_right_eq
tff(fact_6569_mod__mult__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(A2,modulo364778990260209775nteger(B2,C2)),C2) = modulo364778990260209775nteger(times_3573771949741848930nteger(A2,B2),C2) ).

% mod_mult_right_eq
tff(fact_6570_mod__mult__left__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2)),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) ).

% mod_mult_left_eq
tff(fact_6571_mod__mult__left__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A2),C2)),B2)),C2) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A2),B2)),C2) ).

% mod_mult_left_eq
tff(fact_6572_mod__mult__left__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A2,C2),B2),C2) = modulo364778990260209775nteger(times_3573771949741848930nteger(A2,B2),C2) ).

% mod_mult_left_eq
tff(fact_6573_mult__mod__right,axiom,
    ! [C2: nat,A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),B2)) ).

% mult_mod_right
tff(fact_6574_mult__mod__right,axiom,
    ! [C2: int,A2: int,B2: int] : aa_int_int(times_times_int(C2),aa_int_int(modulo_modulo_int(A2),B2)) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(C2),A2)),aa_int_int(times_times_int(C2),B2)) ).

% mult_mod_right
tff(fact_6575_mult__mod__right,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] : times_3573771949741848930nteger(C2,modulo364778990260209775nteger(A2,B2)) = modulo364778990260209775nteger(times_3573771949741848930nteger(C2,A2),times_3573771949741848930nteger(C2,B2)) ).

% mult_mod_right
tff(fact_6576_mod__mult__mult2,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),C2) ).

% mod_mult_mult2
tff(fact_6577_mod__mult__mult2,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A2),C2)),aa_int_int(times_times_int(B2),C2)) = aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A2),B2)),C2) ).

% mod_mult_mult2
tff(fact_6578_mod__mult__mult2,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(A2,C2),times_3573771949741848930nteger(B2,C2)) = times_3573771949741848930nteger(modulo364778990260209775nteger(A2,B2),C2) ).

% mod_mult_mult2
tff(fact_6579_mod__mult__cong,axiom,
    ! [A2: nat,C2: nat,A4: nat,B2: nat,B4: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A4),C2) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),C2) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A4),B4)),C2) ) ) ) ).

% mod_mult_cong
tff(fact_6580_mod__mult__cong,axiom,
    ! [A2: int,C2: int,A4: int,B2: int,B4: int] :
      ( ( aa_int_int(modulo_modulo_int(A2),C2) = aa_int_int(modulo_modulo_int(A4),C2) )
     => ( ( aa_int_int(modulo_modulo_int(B2),C2) = aa_int_int(modulo_modulo_int(B4),C2) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A2),B2)),C2) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A4),B4)),C2) ) ) ) ).

% mod_mult_cong
tff(fact_6581_mod__mult__cong,axiom,
    ! [A2: code_integer,C2: code_integer,A4: code_integer,B2: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A2,C2) = modulo364778990260209775nteger(A4,C2) )
     => ( ( modulo364778990260209775nteger(B2,C2) = modulo364778990260209775nteger(B4,C2) )
       => ( modulo364778990260209775nteger(times_3573771949741848930nteger(A2,B2),C2) = modulo364778990260209775nteger(times_3573771949741848930nteger(A4,B4),C2) ) ) ) ).

% mod_mult_cong
tff(fact_6582_mod__mult__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) ).

% mod_mult_eq
tff(fact_6583_mod__mult__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A2),C2)),aa_int_int(modulo_modulo_int(B2),C2))),C2) = aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A2),B2)),C2) ).

% mod_mult_eq
tff(fact_6584_mod__mult__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : modulo364778990260209775nteger(times_3573771949741848930nteger(modulo364778990260209775nteger(A2,C2),modulo364778990260209775nteger(B2,C2)),C2) = modulo364778990260209775nteger(times_3573771949741848930nteger(A2,B2),C2) ).

% mod_mult_eq
tff(fact_6585_mod__diff__right__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A2),aa_int_int(modulo_modulo_int(B2),C2))),C2) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A2),B2)),C2) ).

% mod_diff_right_eq
tff(fact_6586_mod__diff__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A2,modulo364778990260209775nteger(B2,C2)),C2) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A2,B2),C2) ).

% mod_diff_right_eq
tff(fact_6587_mod__diff__left__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A2),C2)),B2)),C2) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A2),B2)),C2) ).

% mod_diff_left_eq
tff(fact_6588_mod__diff__left__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A2,C2),B2),C2) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A2,B2),C2) ).

% mod_diff_left_eq
tff(fact_6589_mod__diff__cong,axiom,
    ! [A2: int,C2: int,A4: int,B2: int,B4: int] :
      ( ( aa_int_int(modulo_modulo_int(A2),C2) = aa_int_int(modulo_modulo_int(A4),C2) )
     => ( ( aa_int_int(modulo_modulo_int(B2),C2) = aa_int_int(modulo_modulo_int(B4),C2) )
       => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A2),B2)),C2) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A4),B4)),C2) ) ) ) ).

% mod_diff_cong
tff(fact_6590_mod__diff__cong,axiom,
    ! [A2: code_integer,C2: code_integer,A4: code_integer,B2: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A2,C2) = modulo364778990260209775nteger(A4,C2) )
     => ( ( modulo364778990260209775nteger(B2,C2) = modulo364778990260209775nteger(B4,C2) )
       => ( modulo364778990260209775nteger(minus_8373710615458151222nteger(A2,B2),C2) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A4,B4),C2) ) ) ) ).

% mod_diff_cong
tff(fact_6591_mod__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A2),C2)),aa_int_int(modulo_modulo_int(B2),C2))),C2) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(A2),B2)),C2) ).

% mod_diff_eq
tff(fact_6592_mod__diff__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A2,C2),modulo364778990260209775nteger(B2,C2)),C2) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A2,B2),C2) ).

% mod_diff_eq
tff(fact_6593_power__mod,axiom,
    ! [A2: nat,B2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),Nb)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A2),Nb)),B2) ).

% power_mod
tff(fact_6594_power__mod,axiom,
    ! [A2: int,B2: int,Nb: nat] : aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(aa_int_int(modulo_modulo_int(A2),B2)),Nb)),B2) = aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A2),Nb)),B2) ).

% power_mod
tff(fact_6595_power__mod,axiom,
    ! [A2: code_integer,B2: code_integer,Nb: nat] : modulo364778990260209775nteger(power_8256067586552552935nteger(modulo364778990260209775nteger(A2,B2),Nb),B2) = modulo364778990260209775nteger(power_8256067586552552935nteger(A2,Nb),B2) ).

% power_mod
tff(fact_6596_mod__mod__cancel,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2) ) ) ).

% mod_mod_cancel
tff(fact_6597_mod__mod__cancel,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(modulo_modulo_int(A2),B2)),C2) = aa_int_int(modulo_modulo_int(A2),C2) ) ) ).

% mod_mod_cancel
tff(fact_6598_mod__mod__cancel,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),B2)
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(A2,B2),C2) = modulo364778990260209775nteger(A2,C2) ) ) ).

% mod_mod_cancel
tff(fact_6599_dvd__mod,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) ) ) ).

% dvd_mod
tff(fact_6600_dvd__mod,axiom,
    ! [K: int,M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),M)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(modulo_modulo_int(M),Nb)) ) ) ).

% dvd_mod
tff(fact_6601_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,Nb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),modulo364778990260209775nteger(M,Nb)) ) ) ).

% dvd_mod
tff(fact_6602_dvd__mod__imp__dvd,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),A2) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6603_dvd__mod__imp__dvd,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),aa_int_int(modulo_modulo_int(A2),B2))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),A2) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6604_dvd__mod__imp__dvd,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),modulo364778990260209775nteger(A2,B2))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),A2) ) ) ).

% dvd_mod_imp_dvd
tff(fact_6605_dvd__mod__iff,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C2),A2) ) ) ).

% dvd_mod_iff
tff(fact_6606_dvd__mod__iff,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),aa_int_int(modulo_modulo_int(A2),B2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),A2) ) ) ).

% dvd_mod_iff
tff(fact_6607_dvd__mod__iff,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),modulo364778990260209775nteger(A2,B2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),A2) ) ) ).

% dvd_mod_iff
tff(fact_6608_mod__Suc__Suc__eq,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,M))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_6609_mod__Suc__eq,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) ).

% mod_Suc_eq
tff(fact_6610_nat__mod__eq,axiom,
    ! [B2: nat,Nb: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),Nb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Nb) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),Nb) = B2 ) ) ) ).

% nat_mod_eq
tff(fact_6611_mod__plus__right,axiom,
    ! [A2: nat,Xc: nat,M: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),Xc)),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),Xc)),M) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),M) ) ) ).

% mod_plus_right
tff(fact_6612_mod__less__eq__dividend,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)),M) ).

% mod_less_eq_dividend
tff(fact_6613_unset__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),Nb) = aa_int_nat(nat2,aa_int_int(bit_se4203085406695923979it_int(M),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% unset_bit_nat_def
tff(fact_6614_unset__bit__less__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se4203085406695923979it_int(Nb),K)),K) ).

% unset_bit_less_eq
tff(fact_6615_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,modulo364778990260209775nteger(A2,B2)),A2) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_6616_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),A2) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_6617_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(A2),B2)),A2) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_6618_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),B2) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_6619_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A2),B2)),B2) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_6620_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A2,B2)),B2) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_6621_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_6622_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_6623_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q3)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q3)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_6624_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_6625_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) ).

% cong_exp_iff_simps(4)
tff(fact_6626_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(one)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(one)) ).

% cong_exp_iff_simps(4)
tff(fact_6627_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = A2 )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_6628_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A2),B2) = A2 )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_6629_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A2,B2) = A2 )
    <=> ( divide6298287555418463151nteger(A2,B2) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_6630_mod__eqE,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A2),C2) = aa_int_int(modulo_modulo_int(B2),C2) )
     => ~ ! [D5: int] : B2 != aa_int_int(plus_plus_int(A2),aa_int_int(times_times_int(C2),D5)) ) ).

% mod_eqE
tff(fact_6631_mod__eqE,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A2,C2) = modulo364778990260209775nteger(B2,C2) )
     => ~ ! [D5: code_integer] : B2 != plus_p5714425477246183910nteger(A2,times_3573771949741848930nteger(C2,D5)) ) ).

% mod_eqE
tff(fact_6632_div__add1__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2)) ).

% div_add1_eq
tff(fact_6633_div__add1__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(A2),B2)),C2) = aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),C2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A2),C2)),aa_int_int(modulo_modulo_int(B2),C2))),C2)) ).

% div_add1_eq
tff(fact_6634_div__add1__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A2,B2),C2) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A2,C2),divide6298287555418463151nteger(B2,C2)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A2,C2),modulo364778990260209775nteger(B2,C2)),C2)) ).

% div_add1_eq
tff(fact_6635_mod__0__imp__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2) ) ).

% mod_0_imp_dvd
tff(fact_6636_mod__0__imp__dvd,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A2),B2) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2) ) ).

% mod_0_imp_dvd
tff(fact_6637_mod__0__imp__dvd,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A2,B2) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2) ) ).

% mod_0_imp_dvd
tff(fact_6638_dvd__eq__mod__eq__0,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A2),B2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),A2) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_6639_dvd__eq__mod__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A2),B2)
    <=> ( aa_int_int(modulo_modulo_int(B2),A2) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_6640_dvd__eq__mod__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A2),B2)
    <=> ( modulo364778990260209775nteger(B2,A2) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_6641_mod__eq__0__iff__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A2) ) ).

% mod_eq_0_iff_dvd
tff(fact_6642_mod__eq__0__iff__dvd,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A2),B2) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2) ) ).

% mod_eq_0_iff_dvd
tff(fact_6643_mod__eq__0__iff__dvd,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A2,B2) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2) ) ).

% mod_eq_0_iff_dvd
tff(fact_6644_mod__eq__dvd__iff,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( aa_int_int(modulo_modulo_int(A2),C2) = aa_int_int(modulo_modulo_int(B2),C2) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C2),aa_int_int(minus_minus_int(A2),B2)) ) ).

% mod_eq_dvd_iff
tff(fact_6645_mod__eq__dvd__iff,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A2,C2) = modulo364778990260209775nteger(B2,C2) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C2),minus_8373710615458151222nteger(A2,B2)) ) ).

% mod_eq_dvd_iff
tff(fact_6646_dvd__minus__mod,axiom,
    ! [B2: nat,A2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))) ).

% dvd_minus_mod
tff(fact_6647_dvd__minus__mod,axiom,
    ! [B2: int,A2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),aa_int_int(minus_minus_int(A2),aa_int_int(modulo_modulo_int(A2),B2))) ).

% dvd_minus_mod
tff(fact_6648_dvd__minus__mod,axiom,
    ! [B2: code_integer,A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),minus_8373710615458151222nteger(A2,modulo364778990260209775nteger(A2,B2))) ).

% dvd_minus_mod
tff(fact_6649_mod__Suc,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,M)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))) ).

% mod_Suc
tff(fact_6650_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P3: nat,M: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P3)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),P3)
         => ( ! [N: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),P3)
               => ( aa_nat_o(P,N)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N)),P3)) ) )
           => aa_nat_o(P,M) ) ) ) ) ).

% mod_induct
tff(fact_6651_nat__mod__lem,axiom,
    ! [Nb: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),Nb)
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),Nb) = B2 ) ) ) ).

% nat_mod_lem
tff(fact_6652_mod__less__divisor,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_6653_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,M: nat,Nb: nat] :
      ( ! [M4: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M4),zero_zero_nat)
     => ( ! [M4: nat,N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M4),N))
             => aa_nat_o(aa_nat_fun_nat_o(P,M4),N) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,M),Nb) ) ) ).

% gcd_nat_induct
tff(fact_6654_mod__Suc__le__divisor,axiom,
    ! [M: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_6655_word__rot__lem,axiom,
    ! [L: nat,K: nat,D2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,L),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),L)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),L)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,D2),Nb)),L) = Nb ) ) ) ).

% word_rot_lem
tff(fact_6656_nat__minus__mod,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),M))),M) = zero_zero_nat ).

% nat_minus_mod
tff(fact_6657_mod__nat__sub,axiom,
    ! [Xc: nat,Z: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Z)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xc),Ya)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xc),Ya) ) ) ).

% mod_nat_sub
tff(fact_6658_mod__if,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb),M,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb)) ).

% mod_if
tff(fact_6659_mod__geq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_6660_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),D2) = zero_zero_nat )
     => ? [Q5: nat] : M = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D2),Q5) ) ).

% mod_eq_0D
tff(fact_6661_nat__minus__mod__plus__right,axiom,
    ! [Nb: nat,Xc: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),M))),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),M) ).

% nat_minus_mod_plus_right
tff(fact_6662_le__mod__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_6663_msrevs_I2_J,axiom,
    ! [K: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),M)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) ).

% msrevs(2)
tff(fact_6664_nat__mod__eq__iff,axiom,
    ! [Xc: nat,Nb: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ya),Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_6665_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A2,B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_6666_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_6667_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A2),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_6668_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2)
       => ( modulo364778990260209775nteger(A2,B2) = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_6669_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_6670_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
       => ( aa_int_int(modulo_modulo_int(A2),B2) = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_6671_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
tff(fact_6672_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q3: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = zero_zero_int )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
tff(fact_6673_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q3: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
tff(fact_6674_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) = zero_zero_nat ).

% cong_exp_iff_simps(1)
tff(fact_6675_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,one)) = zero_zero_int ).

% cong_exp_iff_simps(1)
tff(fact_6676_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_6677_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_6678_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_6679_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(6)
tff(fact_6680_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_6681_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_6682_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(8)
tff(fact_6683_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6684_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6685_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(10)
tff(fact_6686_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6687_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,Nb: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6688_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) ).

% cong_exp_iff_simps(12)
tff(fact_6689_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6690_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q3)) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6691_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q3)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q3)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_6692_mult__div__mod__eq,axiom,
    ! [B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = A2 ).

% mult_div_mod_eq
tff(fact_6693_mult__div__mod__eq,axiom,
    ! [B2: int,A2: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),aa_int_int(modulo_modulo_int(A2),B2)) = A2 ).

% mult_div_mod_eq
tff(fact_6694_mult__div__mod__eq,axiom,
    ! [B2: code_integer,A2: code_integer] : plus_p5714425477246183910nteger(times_3573771949741848930nteger(B2,divide6298287555418463151nteger(A2,B2)),modulo364778990260209775nteger(A2,B2)) = A2 ).

% mult_div_mod_eq
tff(fact_6695_mod__mult__div__eq,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))) = A2 ).

% mod_mult_div_eq
tff(fact_6696_mod__mult__div__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A2),B2)),aa_int_int(times_times_int(B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))) = A2 ).

% mod_mult_div_eq
tff(fact_6697_mod__mult__div__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A2,B2),times_3573771949741848930nteger(B2,divide6298287555418463151nteger(A2,B2))) = A2 ).

% mod_mult_div_eq
tff(fact_6698_mod__div__mult__eq,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)) = A2 ).

% mod_div_mult_eq
tff(fact_6699_mod__div__mult__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A2),B2)),aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)) = A2 ).

% mod_div_mult_eq
tff(fact_6700_mod__div__mult__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A2,B2),times_3573771949741848930nteger(divide6298287555418463151nteger(A2,B2),B2)) = A2 ).

% mod_div_mult_eq
tff(fact_6701_div__mult__mod__eq,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = A2 ).

% div_mult_mod_eq
tff(fact_6702_div__mult__mod__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)),aa_int_int(modulo_modulo_int(A2),B2)) = A2 ).

% div_mult_mod_eq
tff(fact_6703_div__mult__mod__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A2,B2),B2),modulo364778990260209775nteger(A2,B2)) = A2 ).

% div_mult_mod_eq
tff(fact_6704_mod__div__decomp,axiom,
    ! [A2: nat,B2: nat] : A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) ).

% mod_div_decomp
tff(fact_6705_mod__div__decomp,axiom,
    ! [A2: int,B2: int] : A2 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)),aa_int_int(modulo_modulo_int(A2),B2)) ).

% mod_div_decomp
tff(fact_6706_mod__div__decomp,axiom,
    ! [A2: code_integer,B2: code_integer] : A2 = plus_p5714425477246183910nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A2,B2),B2),modulo364778990260209775nteger(A2,B2)) ).

% mod_div_decomp
tff(fact_6707_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2) ).

% cancel_div_mod_rules(1)
tff(fact_6708_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)),aa_int_int(modulo_modulo_int(A2),B2))),C2) = aa_int_int(plus_plus_int(A2),C2) ).

% cancel_div_mod_rules(1)
tff(fact_6709_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(divide6298287555418463151nteger(A2,B2),B2),modulo364778990260209775nteger(A2,B2)),C2) = plus_p5714425477246183910nteger(A2,C2) ).

% cancel_div_mod_rules(1)
tff(fact_6710_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: nat,A2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),C2) ).

% cancel_div_mod_rules(2)
tff(fact_6711_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: int,A2: int,C2: int] : aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),aa_int_int(modulo_modulo_int(A2),B2))),C2) = aa_int_int(plus_plus_int(A2),C2) ).

% cancel_div_mod_rules(2)
tff(fact_6712_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(B2,divide6298287555418463151nteger(A2,B2)),modulo364778990260209775nteger(A2,B2)),C2) = plus_p5714425477246183910nteger(A2,C2) ).

% cancel_div_mod_rules(2)
tff(fact_6713_div__mult1__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B2)),C2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C2))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),C2))),C2)) ).

% div_mult1_eq
tff(fact_6714_div__mult1__eq,axiom,
    ! [A2: int,B2: int,C2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A2),B2)),C2) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(A2),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C2))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(A2),aa_int_int(modulo_modulo_int(B2),C2))),C2)) ).

% div_mult1_eq
tff(fact_6715_div__mult1__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] : divide6298287555418463151nteger(times_3573771949741848930nteger(A2,B2),C2) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(A2,divide6298287555418463151nteger(B2,C2)),divide6298287555418463151nteger(times_3573771949741848930nteger(A2,modulo364778990260209775nteger(B2,C2)),C2)) ).

% div_mult1_eq
tff(fact_6716_zmde,axiom,
    ! [B2: int,A2: int] : aa_int_int(times_times_int(B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) = aa_int_int(minus_minus_int(A2),aa_int_int(modulo_modulo_int(A2),B2)) ).

% zmde
tff(fact_6717_zmde,axiom,
    ! [B2: code_integer,A2: code_integer] : times_3573771949741848930nteger(B2,divide6298287555418463151nteger(A2,B2)) = minus_8373710615458151222nteger(A2,modulo364778990260209775nteger(A2,B2)) ).

% zmde
tff(fact_6718_minus__mult__div__eq__mod,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% minus_mult_div_eq_mod
tff(fact_6719_minus__mult__div__eq__mod,axiom,
    ! [A2: int,B2: int] : aa_int_int(minus_minus_int(A2),aa_int_int(times_times_int(B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))) = aa_int_int(modulo_modulo_int(A2),B2) ).

% minus_mult_div_eq_mod
tff(fact_6720_minus__mult__div__eq__mod,axiom,
    ! [A2: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A2,times_3573771949741848930nteger(B2,divide6298287555418463151nteger(A2,B2))) = modulo364778990260209775nteger(A2,B2) ).

% minus_mult_div_eq_mod
tff(fact_6721_minus__mod__eq__mult__div,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)) ).

% minus_mod_eq_mult_div
tff(fact_6722_minus__mod__eq__mult__div,axiom,
    ! [A2: int,B2: int] : aa_int_int(minus_minus_int(A2),aa_int_int(modulo_modulo_int(A2),B2)) = aa_int_int(times_times_int(B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ).

% minus_mod_eq_mult_div
tff(fact_6723_minus__mod__eq__mult__div,axiom,
    ! [A2: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A2,modulo364778990260209775nteger(A2,B2)) = times_3573771949741848930nteger(B2,divide6298287555418463151nteger(A2,B2)) ).

% minus_mod_eq_mult_div
tff(fact_6724_minus__mod__eq__div__mult,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_6725_minus__mod__eq__div__mult,axiom,
    ! [A2: int,B2: int] : aa_int_int(minus_minus_int(A2),aa_int_int(modulo_modulo_int(A2),B2)) = aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_6726_minus__mod__eq__div__mult,axiom,
    ! [A2: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A2,modulo364778990260209775nteger(A2,B2)) = times_3573771949741848930nteger(divide6298287555418463151nteger(A2,B2),B2) ).

% minus_mod_eq_div_mult
tff(fact_6727_minus__div__mult__eq__mod,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ).

% minus_div_mult_eq_mod
tff(fact_6728_minus__div__mult__eq__mod,axiom,
    ! [A2: int,B2: int] : aa_int_int(minus_minus_int(A2),aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2)) = aa_int_int(modulo_modulo_int(A2),B2) ).

% minus_div_mult_eq_mod
tff(fact_6729_minus__div__mult__eq__mod,axiom,
    ! [A2: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A2,times_3573771949741848930nteger(divide6298287555418463151nteger(A2,B2),B2)) = modulo364778990260209775nteger(A2,B2) ).

% minus_div_mult_eq_mod
tff(fact_6730_unit__imp__mod__eq__0,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_6731_unit__imp__mod__eq__0,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_int(modulo_modulo_int(A2),B2) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_6732_unit__imp__mod__eq__0,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A2,B2) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_6733_mod__le__divisor,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_6734_div__less__mono,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Nb) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Nb) = zero_zero_nat )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_6735_mod__nat__add,axiom,
    ! [Xc: nat,Z: nat,Ya: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ya),Z)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Ya)),Z) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Ya)),Z),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Ya),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xc),Ya)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_6736_nat__mod__eq__lemma,axiom,
    ! [Xc: nat,Nb: nat,Ya: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Ya),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),Xc)
       => ? [Q5: nat] : Xc = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Ya),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q5)) ) ) ).

% nat_mod_eq_lemma
tff(fact_6737_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => ~ ! [S3: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_6738_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
       => ~ ! [S3: nat] : M != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q3),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_6739_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_6740_divmod_H__nat__def,axiom,
    ! [M: num,Nb: num] : unique5055182867167087721od_nat(M,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_6741_mod__mult2__eq,axiom,
    ! [M: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),Q3))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) ).

% mod_mult2_eq
tff(fact_6742_div__mod__decomp,axiom,
    ! [A: nat,Nb: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),Nb)) ).

% div_mod_decomp
tff(fact_6743_modulo__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_6744_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,M: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),Q3) )
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_6745_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Vaa: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] : vEBT_VEBT_minNulli(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz2),Vaa,Vb2),Vc2)) = heap_Time_return_o($false) ).

% VEBT_internal.minNulli.simps(5)
tff(fact_6746_VEBT__internal_OminNulli_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] : vEBT_VEBT_minNulli(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw,Ux2),Uy2)) = heap_Time_return_o($true) ).

% VEBT_internal.minNulli.simps(4)
tff(fact_6747_vebt__buildupi_Osimps_I1_J,axiom,
    vEBT_vebt_buildupi(zero_zero_nat) = heap_T3630416162098727440_VEBTi(aa_o_VEBT_VEBTi(vEBT_Leafi($false),$false)) ).

% vebt_buildupi.simps(1)
tff(fact_6748_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
    vEBT_V739175172307565963ildupi(zero_zero_nat) = heap_T3630416162098727440_VEBTi(aa_o_VEBT_VEBTi(vEBT_Leafi($false),$false)) ).

% VEBT_internal.vebt_buildupi'.simps(1)
tff(fact_6749_star__assoc,axiom,
    ! [A2: assn,B2: assn,C2: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),B2)),C2) = aa_assn_assn(times_times_assn(A2),aa_assn_assn(times_times_assn(B2),C2)) ).

% star_assoc
tff(fact_6750_star__aci_I2_J,axiom,
    ! [A2: assn,B2: assn] : aa_assn_assn(times_times_assn(A2),B2) = aa_assn_assn(times_times_assn(B2),A2) ).

% star_aci(2)
tff(fact_6751_star__aci_I3_J,axiom,
    ! [A2: assn,B2: assn,C2: assn] : aa_assn_assn(times_times_assn(A2),aa_assn_assn(times_times_assn(B2),C2)) = aa_assn_assn(times_times_assn(B2),aa_assn_assn(times_times_assn(A2),C2)) ).

% star_aci(3)
tff(fact_6752_assn__aci_I10_J,axiom,
    ! [A2: assn,B2: assn,C2: assn] : aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),B2)),C2) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A2),C2)),B2) ).

% assn_aci(10)
tff(fact_6753_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
     => entails(P,Q) ) ).

% is_entails
tff(fact_6754_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q3: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) != zero_zero_nat ).

% cong_exp_iff_simps(3)
tff(fact_6755_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q3: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) != zero_zero_int ).

% cong_exp_iff_simps(3)
tff(fact_6756_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q3: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) != zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(3)
tff(fact_6757_mod__mult2__eq_H,axiom,
    ! [A2: int,M: nat,Nb: nat] : aa_int_int(modulo_modulo_int(A2),aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(semiri1314217659103216013at_int,M))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(modulo_modulo_int(A2),aa_nat_int(semiri1314217659103216013at_int,M))) ).

% mod_mult2_eq'
tff(fact_6758_mod__mult2__eq_H,axiom,
    ! [A2: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,M)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(semiri1316708129612266289at_nat,M))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(semiri1316708129612266289at_nat,M))) ).

% mod_mult2_eq'
tff(fact_6759_mod__mult2__eq_H,axiom,
    ! [A2: code_integer,M: nat,Nb: nat] : modulo364778990260209775nteger(A2,times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,M),modulo364778990260209775nteger(divide6298287555418463151nteger(A2,aa_nat_Code_integer(semiri4939895301339042750nteger,M)),aa_nat_Code_integer(semiri4939895301339042750nteger,Nb))),modulo364778990260209775nteger(A2,aa_nat_Code_integer(semiri4939895301339042750nteger,M))) ).

% mod_mult2_eq'
tff(fact_6760_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))))) ) ).

% even_even_mod_4_iff
tff(fact_6761_unset__bit__Suc,axiom,
    ! [Nb: nat,A2: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,Nb),A2) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_se8260200283734997820nteger(Nb,divide6298287555418463151nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6762_unset__bit__Suc,axiom,
    ! [Nb: nat,A2: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,Nb)),A2) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6763_unset__bit__Suc,axiom,
    ! [Nb: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,Nb)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_6764_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,M)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)))),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_6765_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,M)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_6766_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,M)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)))),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_6767_mod__lemma,axiom,
    ! [C2: nat,R3: nat,B2: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R3),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Q3),C2))),R3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) ) ) ).

% mod_lemma
tff(fact_6768_split__mod,axiom,
    ! [P: fun_nat_o,M: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,M) )
        & ( ( Nb != zero_zero_nat )
         => ! [I2: nat,J: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),Nb)
             => ( ( M = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I2)),J) )
               => aa_nat_o(P,J) ) ) ) ) ) ).

% split_mod
tff(fact_6769_divmod__def,axiom,
    ! [M: num,Nb: num] : unique5055182867167087721od_nat(M,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_6770_divmod__def,axiom,
    ! [M: num,Nb: num] : unique5052692396658037445od_int(M,Nb) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_def
tff(fact_6771_divmod__def,axiom,
    ! [M: num,Nb: num] : unique3479559517661332726nteger(M,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Nb)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Nb))) ).

% divmod_def
tff(fact_6772_diff__mod__le,axiom,
    ! [A2: nat,D2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),D2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,D2),B2)) ) ) ).

% diff_mod_le
tff(fact_6773_mod__nat__eqI,axiom,
    ! [R3: nat,Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R3),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R3),M)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),R3))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb) = R3 ) ) ) ) ).

% mod_nat_eqI
tff(fact_6774_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    vEBT_V739175172307565963ildupi(aa_nat_nat(suc,zero_zero_nat)) = heap_T3630416162098727440_VEBTi(aa_o_VEBT_VEBTi(vEBT_Leafi($false),$false)) ).

% VEBT_internal.vebt_buildupi'.simps(2)
tff(fact_6775_vebt__buildupi_Osimps_I2_J,axiom,
    vEBT_vebt_buildupi(aa_nat_nat(suc,zero_zero_nat)) = heap_T3630416162098727440_VEBTi(aa_o_VEBT_VEBTi(vEBT_Leafi($false),$false)) ).

% vebt_buildupi.simps(2)
tff(fact_6776_real__of__nat__div__aux,axiom,
    ! [Xc: nat,D2: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Xc)),aa_nat_real(semiri5074537144036343181t_real,D2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xc),D2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),D2))),aa_nat_real(semiri5074537144036343181t_real,D2))) ).

% real_of_nat_div_aux
tff(fact_6777_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : vEBT_vebt_maxti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv2),Uw)) = heap_T3487192422709364219on_nat(none_nat) ).

% vebt_maxti.simps(2)
tff(fact_6778_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: array_VEBT_VEBTi,Uw: vEBT_VEBTi] : vEBT_vebt_minti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu,Uv2),Uw)) = heap_T3487192422709364219on_nat(none_nat) ).

% vebt_minti.simps(2)
tff(fact_6779_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_6780_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C2)
     => ( modulo364778990260209775nteger(A2,times_3573771949741848930nteger(B2,C2)) = plus_p5714425477246183910nteger(times_3573771949741848930nteger(B2,modulo364778990260209775nteger(divide6298287555418463151nteger(A2,B2),C2)),modulo364778990260209775nteger(A2,B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_6781_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),C2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2)),C2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_6782_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(modulo_modulo_int(A2),aa_int_int(times_times_int(B2),C2)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2))),aa_int_int(modulo_modulo_int(A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_6783_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Nb: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6784_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Nb: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6785_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
tff(fact_6786_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,M))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,one)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Q3))) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,M)),aa_num_nat(numeral_numeral_nat,Q3)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6787_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) = aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q3))) )
    <=> ( aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Q3)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6788_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,M)),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q3))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Q3)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_6789_even__iff__mod__2__eq__zero,axiom,
    ! [A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6790_even__iff__mod__2__eq__zero,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
    <=> ( aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6791_even__iff__mod__2__eq__zero,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
    <=> ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_6792_odd__iff__mod__2__eq__one,axiom,
    ! [A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6793_odd__iff__mod__2__eq__one,axiom,
    ! [A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
    <=> ( aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6794_odd__iff__mod__2__eq__one,axiom,
    ! [A2: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
    <=> ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_6795_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,M)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),M)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_6796_Suc__times__mod__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb))),M) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_6797_VEBT__internal_OminNulli_Oelims,axiom,
    ! [Xc: vEBT_VEBTi,Ya: heap_Time_Heap_o] :
      ( ( vEBT_VEBT_minNulli(Xc) = Ya )
     => ( ( ( Xc = aa_o_VEBT_VEBTi(vEBT_Leafi($false),$false) )
         => ( Ya != heap_Time_return_o($true) ) )
       => ( ( ? [Uv: $o] : Xc = aa_o_VEBT_VEBTi(vEBT_Leafi($true),(Uv))
           => ( Ya != heap_Time_return_o($false) ) )
         => ( ( ? [Uu2: $o] : Xc = aa_o_VEBT_VEBTi(vEBT_Leafi((Uu2)),$true)
             => ( Ya != heap_Time_return_o($false) ) )
           => ( ( ? [Uw2: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] : Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux),Uy)
               => ( Ya != heap_Time_return_o($true) ) )
             => ~ ( ? [Uz: product_prod_nat_nat,Va: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] : Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc)
                 => ( Ya != heap_Time_return_o($false) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.elims
tff(fact_6798_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : vEBT_vebt_maxti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Ux2,Uy2),Uz2)) = heap_T3487192422709364219on_nat(some_nat(Maa)) ).

% vebt_maxti.simps(3)
tff(fact_6799_vebt__minti_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] : vEBT_vebt_minti(aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Ux2,Uy2),Uz2)) = heap_T3487192422709364219on_nat(some_nat(Mia)) ).

% vebt_minti.simps(3)
tff(fact_6800_divmod__digit__0_I2_J,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_6801_divmod__digit__0_I2_J,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A2),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_int_int(modulo_modulo_int(A2),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)) = aa_int_int(modulo_modulo_int(A2),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_6802_divmod__digit__0_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2))),B2)
       => ( modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2)) = modulo364778990260209775nteger(A2,B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_6803_bits__stable__imp__add__self,axiom,
    ! [A2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A2 )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
tff(fact_6804_bits__stable__imp__add__self,axiom,
    ! [A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A2 )
     => ( aa_int_int(plus_plus_int(A2),aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_6805_bits__stable__imp__add__self,axiom,
    ! [A2: code_integer] :
      ( ( divide6298287555418463151nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A2 )
     => ( plus_p5714425477246183910nteger(A2,modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))) = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
tff(fact_6806_parity__cases,axiom,
    ! [A2: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_6807_parity__cases,axiom,
    ! [A2: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
       => ( aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
         => ( aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_6808_parity__cases,axiom,
    ! [A2: code_integer] :
      ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
       => ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2)
         => ( modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_6809_mod2__eq__if,axiom,
    ! [A2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_6810_mod2__eq__if,axiom,
    ! [A2: int] :
      aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_6811_mod2__eq__if,axiom,
    ! [A2: code_integer] :
      modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_6812_div__exp__mod__exp__eq,axiom,
    ! [A2: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_6813_div__exp__mod__exp__eq,axiom,
    ! [A2: int,Nb: nat,M: nat] : aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(modulo_modulo_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_6814_div__exp__mod__exp__eq,axiom,
    ! [A2: code_integer,Nb: nat,M: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A2,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A2,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M))),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_6815_power__mod__div,axiom,
    ! [Xc: nat,Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))) ).

% power_mod_div
tff(fact_6816_verit__le__mono__div,axiom,
    ! [A: nat,B: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,
              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_6817_vebt__maxti_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_maxti(aa_o_VEBT_VEBTi(vEBT_Leafi((A2)),(B2))) = $ite(
        (B2),
        heap_T3487192422709364219on_nat(some_nat(one_one_nat)),
        $ite((A2),heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ).

% vebt_maxti.simps(1)
tff(fact_6818_vebt__minti_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_minti(aa_o_VEBT_VEBTi(vEBT_Leafi((A2)),(B2))) = $ite(
        (A2),
        heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),
        $ite((B2),heap_T3487192422709364219on_nat(some_nat(one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ).

% vebt_minti.simps(1)
tff(fact_6819_divmod__digit__0_I1_J,axiom,
    ! [B2: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_6820_divmod__digit__0_I1_J,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A2),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_6821_divmod__digit__0_I1_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2))),B2)
       => ( times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2))) = divide6298287555418463151nteger(A2,B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_6822_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Nb: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6823_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Nb: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6824_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,Nb: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( modulo364778990260209775nteger(times_3573771949741848930nteger(A2,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = times_3573771949741848930nteger(modulo364778990260209775nteger(A2,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M))),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_6825_mod__double__modulus,axiom,
    ! [M: code_integer,Xc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),M)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xc)
       => ( ( modulo364778990260209775nteger(Xc,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M)) = modulo364778990260209775nteger(Xc,M) )
          | ( modulo364778990260209775nteger(Xc,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),M)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Xc,M),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6826_mod__double__modulus,axiom,
    ! [M: nat,Xc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xc)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),M) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6827_mod__double__modulus,axiom,
    ! [M: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
       => ( ( aa_int_int(modulo_modulo_int(Xc),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(modulo_modulo_int(Xc),M) )
          | ( aa_int_int(modulo_modulo_int(Xc),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),M)) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(Xc),M)),M) ) ) ) ) ).

% mod_double_modulus
tff(fact_6828_divmod__digit__1_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2)),B2) = modulo364778990260209775nteger(A2,B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_6829_divmod__digit__1_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_6830_divmod__digit__1_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(modulo_modulo_int(A2),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))
         => ( aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A2),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2) = aa_int_int(modulo_modulo_int(A2),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_6831_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F: assn,Q: assn] :
      ( entails(P,R)
     => ( entails(Ps,aa_assn_assn(times_times_assn(P),F))
       => ( entails(aa_assn_assn(times_times_assn(R),F),Q)
         => entails(Ps,Q) ) ) ) ).

% ent_frame_fwd
tff(fact_6832_fr__rot__rhs,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( entails(A,aa_assn_assn(times_times_assn(B),C))
     => entails(A,aa_assn_assn(times_times_assn(C),B)) ) ).

% fr_rot_rhs
tff(fact_6833_fr__refl,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( entails(A,B)
     => entails(aa_assn_assn(times_times_assn(A),C),aa_assn_assn(times_times_assn(B),C)) ) ).

% fr_refl
tff(fact_6834_fr__rot,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( entails(aa_assn_assn(times_times_assn(A),B),C)
     => entails(aa_assn_assn(times_times_assn(B),A),C) ) ).

% fr_rot
tff(fact_6835_eq__diff__eq_H,axiom,
    ! [Xc: real,Ya: real,Z: real] :
      ( ( Xc = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),Z) )
    <=> ( Ya = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Z) ) ) ).

% eq_diff_eq'
tff(fact_6836_set__bit__Suc,axiom,
    ! [Nb: nat,A2: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,Nb),A2) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_se2793503036327961859nteger(Nb,divide6298287555418463151nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_6837_set__bit__Suc,axiom,
    ! [Nb: nat,A2: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,Nb)),A2) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se7879613467334960850it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_6838_set__bit__Suc,axiom,
    ! [Nb: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,Nb)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_6839_norm__assertion__simps_I1_J,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(one_one_assn),A2) = A2 ).

% norm_assertion_simps(1)
tff(fact_6840_norm__assertion__simps_I2_J,axiom,
    ! [A2: assn] : aa_assn_assn(times_times_assn(A2),one_one_assn) = A2 ).

% norm_assertion_simps(2)
tff(fact_6841_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(suc,zero_zero_nat) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% even_mod_4_div_2
tff(fact_6842_even__unset__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se4203085406695923979it_int(M),A2))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6843_even__unset__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,M),A2))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_6844_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
     => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% odd_mod_4_div_2
tff(fact_6845_frame__rule__left,axiom,
    ! [P: assn,C2: heap_Time_Heap_o,Q: fun_o_assn,R: assn] :
      ( hoare_hoare_triple_o(P,C2,Q)
     => hoare_hoare_triple_o(aa_assn_assn(times_times_assn(R),P),C2,aa_assn_fun_o_assn(aTP_Lamm_em(Q),R)) ) ).

% frame_rule_left
tff(fact_6846_frame__rule__left,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: fun_VEBT_VEBTi_assn,R: assn] :
      ( hoare_1429296392585015714_VEBTi(P,C2,Q)
     => hoare_1429296392585015714_VEBTi(aa_assn_assn(times_times_assn(R),P),C2,aa_ass4389913640333755968i_assn(aTP_Lamm_en(Q),R)) ) ).

% frame_rule_left
tff(fact_6847_frame__rule__left,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: fun_option_nat_assn,R: assn] :
      ( hoare_7629718768684598413on_nat(P,C2,Q)
     => hoare_7629718768684598413on_nat(aa_assn_assn(times_times_assn(R),P),C2,aa_ass4112278883933122603t_assn(aTP_Lamm_eo(Q),R)) ) ).

% frame_rule_left
tff(fact_6848_frame__rule__left,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: fun_nat_assn,R: assn] :
      ( hoare_3067605981109127869le_nat(P,C2,Q)
     => hoare_3067605981109127869le_nat(aa_assn_assn(times_times_assn(R),P),C2,aa_assn_fun_nat_assn(aTP_Lamm_ep(Q),R)) ) ).

% frame_rule_left
tff(fact_6849_divmod__digit__1_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),modulo364778990260209775nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2)))
         => ( plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),divide6298287555418463151nteger(A2,times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),B2))),one_one_Code_integer) = divide6298287555418463151nteger(A2,B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_6850_divmod__digit__1_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_6851_divmod__digit__1_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(modulo_modulo_int(A2),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))
         => ( aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_6852_vebt__maxti_Oelims,axiom,
    ! [Xc: vEBT_VEBTi,Ya: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_maxti(Xc) = Ya )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBTi(vEBT_Leafi((A3)),(B3)) )
           => ( Ya != $ite(
                  (B3),
                  heap_T3487192422709364219on_nat(some_nat(one_one_nat)),
                  $ite((A3),heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
           => ( Ya != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)
               => ( Ya != heap_T3487192422709364219on_nat(some_nat(Ma)) ) ) ) ) ) ).

% vebt_maxti.elims
tff(fact_6853_vebt__minti_Oelims,axiom,
    ! [Xc: vEBT_VEBTi,Ya: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_minti(Xc) = Ya )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBTi(vEBT_Leafi((A3)),(B3)) )
           => ( Ya != $ite(
                  (A3),
                  heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),
                  $ite((B3),heap_T3487192422709364219on_nat(some_nat(one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] : Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)
           => ( Ya != heap_T3487192422709364219on_nat(none_nat) ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] : Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)
               => ( Ya != heap_T3487192422709364219on_nat(some_nat(Mi)) ) ) ) ) ) ).

% vebt_minti.elims
tff(fact_6854_mod__frame__fwd,axiom,
    ! [Ps: assn,H: produc3658429121746597890et_nat,P: assn,R: assn,F: assn] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(Ps),H)
     => ( entails(P,R)
       => ( entails(Ps,aa_assn_assn(times_times_assn(P),F))
         => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(R),F)),H) ) ) ) ).

% mod_frame_fwd
tff(fact_6855_div__half__nat,axiom,
    ! [Ya: nat,Xc: nat] :
      ( ( Ya != zero_zero_nat )
     => ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xc),Ya)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xc),Ya)) = $let(
            q: nat,
            q:= aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Ya)),
            $let(
              r: nat,
              r:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xc),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,q),Ya)),
              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ya),r),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,q),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,r),Ya)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(q),r)) ) ) ) ) ).

% div_half_nat
tff(fact_6856_flip__bit__Suc,axiom,
    ! [Nb: nat,A2: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,Nb),A2) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_se1345352211410354436nteger(Nb,divide6298287555418463151nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_6857_flip__bit__Suc,axiom,
    ! [Nb: nat,A2: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Nb)),A2) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_6858_flip__bit__Suc,axiom,
    ! [Nb: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,Nb)),A2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_6859_tanh__ln__real,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_6860_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(plus_plus_int(A2),one_one_int),B2),product_Pair_int_int(Q3,R3))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),product_Pair_int_int(Q3,aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R3)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_6861_product__nth,axiom,
    ! [Nb: nat,Xs: list_Code_integer,Ys: list_Code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s3445333598471063425nteger(Xs)),size_s3445333598471063425nteger(Ys)))
     => ( nth_Pr2304437835452373666nteger(produc8792966785426426881nteger(Xs,Ys),Nb) = produc1086072967326762835nteger(nth_Code_integer(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s3445333598471063425nteger(Ys))),nth_Code_integer(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s3445333598471063425nteger(Ys)))) ) ) ).

% product_nth
tff(fact_6862_product__nth,axiom,
    ! [Nb: nat,Xs: list_num,Ys: list_num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_num(Xs)),size_size_list_num(Ys)))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys),Nb) = product_Pair_num_num(aa_nat_num(nth_num(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys))),aa_nat_num(nth_num(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys)))) ) ) ).

% product_nth
tff(fact_6863_product__nth,axiom,
    ! [Nb: nat,Xs: list_int,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys),Nb) = product_Pair_int_int(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_6864_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys),Nb) = produc537772716801021591T_VEBT(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_6865_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_real(Ys)))
     => ( nth_Pr6842391030413306568T_real(produc4908677263432625371T_real(Xs,Ys),Nb) = produc8117437818029410057T_real(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_real(Ys))),aa_nat_real(nth_real(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_real(Ys)))) ) ) ).

% product_nth
tff(fact_6866_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys),Nb) = produc8721562602347293563VEBT_o(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys))),aa_nat_o(nth_o(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys)))) ) ) ).

% product_nth
tff(fact_6867_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys),Nb) = produc738532404422230701BT_nat(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys))),aa_nat_nat(nth_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys)))) ) ) ).

% product_nth
tff(fact_6868_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s7982070591426661849_VEBTi(Ys)))
     => ( nth_Pr316670251186196177_VEBTi(produc316462671093861988_VEBTi(Xs,Ys),Nb) = produc6084888613844515218_VEBTi(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s7982070591426661849_VEBTi(Ys))),nth_VEBT_VEBTi(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s7982070591426661849_VEBTi(Ys)))) ) ) ).

% product_nth
tff(fact_6869_product__nth,axiom,
    ! [Nb: nat,Xs: list_real,Ys: list_VEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_s6755466524823107622T_VEBT(Ys)))
     => ( nth_Pr5429231388385915574T_VEBT(produc3722688996059531265T_VEBT(Xs,Ys),Nb) = produc6931449550656315951T_VEBT(aa_nat_real(nth_real(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys))),aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys)))) ) ) ).

% product_nth
tff(fact_6870_product__nth,axiom,
    ! [Nb: nat,Xs: list_real,Ys: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_real(Ys)))
     => ( nth_Pr7489471911767062336l_real(product_real_real(Xs,Ys),Nb) = produc4511245868158468465l_real(aa_nat_real(nth_real(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_real(Ys))),aa_nat_real(nth_real(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_real(Ys)))) ) ) ).

% product_nth
tff(fact_6871_vebt__assn__raw_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: vEBT_VEBTi,Ya: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xc),Xaa) = Ya )
     => ( ! [A3: $o,B3: $o] :
            ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xaa = aa_o_VEBT_VEBTi(vEBT_Leafi((Ai)),(Bi)) )
               => ( Ya != aa_o_assn(pure_assn,
                      ( ( (Ai) = (A3) )
                      & ( (Bi) = (B3) ) )) ) ) )
       => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary) )
             => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                  ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi) )
                 => ( Ya != aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_o_assn(pure_assn,
                                ( ( Mmoi = Mmo )
                                & ( Degi = Deg2 ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Summaryi))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_eq(Tree_list),Tree_array))) ) ) )
         => ( ( ? [V2: option4927543243414619207at_nat,Va2: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] : Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3)
             => ( ? [Vd3: $o,Ve3: $o] : Xaa = aa_o_VEBT_VEBTi(vEBT_Leafi((Vd3)),(Ve3))
               => ( Ya != bot_bot_assn ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Vd3)),(Ve3))
               => ( ? [V2: option4927543243414619207at_nat,Va2: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] : Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3)
                 => ( Ya != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
tff(fact_6872_ex__assn__const,axiom,
    ! [C2: assn] : ex_ass463751140784270563_VEBTi(aTP_Lamm_er(C2)) = C2 ).

% ex_assn_const
tff(fact_6873_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2159334234014336723it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_6874_flip__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2159334234014336723it_int(Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% flip_bit_negative_int_iff
tff(fact_6875_tanh__0,axiom,
    aa_complex_complex(tanh_complex,zero_zero_complex) = zero_zero_complex ).

% tanh_0
tff(fact_6876_tanh__0,axiom,
    aa_real_real(tanh_real,zero_zero_real) = zero_zero_real ).

% tanh_0
tff(fact_6877_tanh__real__zero__iff,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(tanh_real,Xc) = zero_zero_real )
    <=> ( Xc = zero_zero_real ) ) ).

% tanh_real_zero_iff
tff(fact_6878_tanh__real__less__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xc)),aa_real_real(tanh_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ).

% tanh_real_less_iff
tff(fact_6879_tanh__real__le__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xc)),aa_real_real(tanh_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ).

% tanh_real_le_iff
tff(fact_6880_norm__assertion__simps_I17_J,axiom,
    ! [R: assn,Q: fun_li3352006551604695530i_assn] : aa_assn_assn(times_times_assn(R),ex_ass463751140784270563_VEBTi(Q)) = ex_ass463751140784270563_VEBTi(aa_fun6795544665460849810i_assn(aTP_Lamm_es(R),Q)) ).

% norm_assertion_simps(17)
tff(fact_6881_norm__assertion__simps_I16_J,axiom,
    ! [Q: fun_li3352006551604695530i_assn,R: assn] : aa_assn_assn(times_times_assn(ex_ass463751140784270563_VEBTi(Q)),R) = ex_ass463751140784270563_VEBTi(aa_ass1897699608592665680i_assn(aTP_Lamm_et(Q),R)) ).

% norm_assertion_simps(16)
tff(fact_6882_triv__exI,axiom,
    ! [Q: fun_li3352006551604695530i_assn,Xc: list_VEBT_VEBTi] : entails(aa_lis2508312872377092907i_assn(Q,Xc),ex_ass463751140784270563_VEBTi(Q)) ).

% triv_exI
tff(fact_6883_mod__ex__dist,axiom,
    ! [P: fun_li3352006551604695530i_assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(ex_ass463751140784270563_VEBTi(P)),H)
    <=> ? [X2: list_VEBT_VEBTi] : aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(P,X2)),H) ) ).

% mod_ex_dist
tff(fact_6884_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_6885_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(modulo_modulo_int(K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_6886_tanh__real__neg__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xc)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_6887_tanh__real__pos__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc) ) ).

% tanh_real_pos_iff
tff(fact_6888_tanh__real__nonneg__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc) ) ).

% tanh_real_nonneg_iff
tff(fact_6889_tanh__real__nonpos__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xc)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_6890_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_6891_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real] : size_s5035110155006384947T_real(produc4908677263432625371T_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6892_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o] : size_s9168528473962070013VEBT_o(product_VEBT_VEBT_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_6893_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_6894_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] : size_s3387956428969716412_VEBTi(produc316462671093861988_VEBTi(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s7982070591426661849_VEBTi(Ys)) ).

% length_product
tff(fact_6895_length__product,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT] : size_s3289364478449617953T_VEBT(produc3722688996059531265T_VEBT(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_s6755466524823107622T_VEBT(Ys)) ).

% length_product
tff(fact_6896_length__product,axiom,
    ! [Xs: list_real,Ys: list_real] : size_s3932428310213730859l_real(product_real_real(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_real(Ys)) ).

% length_product
tff(fact_6897_length__product,axiom,
    ! [Xs: list_real,Ys: list_o] : size_s987546567493390085real_o(product_real_o(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_o(Ys)) ).

% length_product
tff(fact_6898_length__product,axiom,
    ! [Xs: list_real,Ys: list_nat] : size_s1877336372972134351al_nat(product_real_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_6899_length__product,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBTi] : size_s6963394564282275252_VEBTi(produc5599769701989005224_VEBTi(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_real(Xs)),size_s7982070591426661849_VEBTi(Ys)) ).

% length_product
tff(fact_6900_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_6901_mod__h__bot__iff_I8_J,axiom,
    ! [R: fun_li3352006551604695530i_assn,H: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(ex_ass463751140784270563_VEBTi(R)),produc7507926704131184380et_nat(H,bot_bot_set_nat))
    <=> ? [X2: list_VEBT_VEBTi] : aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(R,X2)),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ) ).

% mod_h_bot_iff(8)
tff(fact_6902_one__mod__exp__eq__one,axiom,
    ! [Nb: nat] : aa_int_int(modulo_modulo_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) = one_one_int ).

% one_mod_exp_eq_one
tff(fact_6903_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_6904_eucl__rel__int,axiom,
    ! [K: int,L: int] : aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L),aa_int_int(modulo_modulo_int(K),L))) ).

% eucl_rel_int
tff(fact_6905_zmod__helper,axiom,
    ! [Nb: int,M: int,K: int,A2: int] :
      ( ( aa_int_int(modulo_modulo_int(Nb),M) = K )
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(Nb),A2)),M) = aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(K),A2)),M) ) ) ).

% zmod_helper
tff(fact_6906_Word_Omod__minus__cong,axiom,
    ! [B2: int,B4: int,Xc: int,X5: int,Ya: int,Y5: int,Z5: int] :
      ( ( B2 = B4 )
     => ( ( aa_int_int(modulo_modulo_int(Xc),B4) = aa_int_int(modulo_modulo_int(X5),B4) )
       => ( ( aa_int_int(modulo_modulo_int(Ya),B4) = aa_int_int(modulo_modulo_int(Y5),B4) )
         => ( ( aa_int_int(minus_minus_int(X5),Y5) = Z5 )
           => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(Xc),Ya)),B2) = aa_int_int(modulo_modulo_int(Z5),B4) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_6907_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(Q3,R3))
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = Q3 ) ) ).

% div_int_unique
tff(fact_6908_ex__distrib__star,axiom,
    ! [P: fun_li3352006551604695530i_assn,Q: assn] : ex_ass463751140784270563_VEBTi(aa_ass1897699608592665680i_assn(aTP_Lamm_et(P),Q)) = aa_assn_assn(times_times_assn(ex_ass463751140784270563_VEBTi(P)),Q) ).

% ex_distrib_star
tff(fact_6909_ent__ex__preI,axiom,
    ! [P: fun_li3352006551604695530i_assn,Q: assn] :
      ( ! [X3: list_VEBT_VEBTi] : entails(aa_lis2508312872377092907i_assn(P,X3),Q)
     => entails(ex_ass463751140784270563_VEBTi(P),Q) ) ).

% ent_ex_preI
tff(fact_6910_ent__ex__postI,axiom,
    ! [P: assn,Q: fun_li3352006551604695530i_assn,Xc: list_VEBT_VEBTi] :
      ( entails(P,aa_lis2508312872377092907i_assn(Q,Xc))
     => entails(P,ex_ass463751140784270563_VEBTi(Q)) ) ).

% ent_ex_postI
tff(fact_6911_enorm__exI_H,axiom,
    ! [Z6: fun_li984516689242532972EBTi_o,P: assn,Q: fun_li3352006551604695530i_assn] :
      ( ! [X3: list_VEBT_VEBTi] :
          ( aa_list_VEBT_VEBTi_o(Z6,X3)
         => entails(P,aa_lis2508312872377092907i_assn(Q,X3)) )
     => ( ? [X_13: list_VEBT_VEBTi] : aa_list_VEBT_VEBTi_o(Z6,X_13)
       => entails(P,ex_ass463751140784270563_VEBTi(Q)) ) ) ).

% enorm_exI'
tff(fact_6912_ex__one__point__gen,axiom,
    ! [P: fun_li3352006551604695530i_assn,V: list_VEBT_VEBTi] :
      ( ! [H4: produc3658429121746597890et_nat,X3: list_VEBT_VEBTi] :
          ( aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(P,X3)),H4)
         => ( X3 = V ) )
     => ( ex_ass463751140784270563_VEBTi(P) = aa_lis2508312872377092907i_assn(P,V) ) ) ).

% ex_one_point_gen
tff(fact_6913_mod__exI,axiom,
    ! [P: fun_li3352006551604695530i_assn,H: produc3658429121746597890et_nat] :
      ( ? [X4: list_VEBT_VEBTi] : aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(P,X4)),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(ex_ass463751140784270563_VEBTi(P)),H) ) ).

% mod_exI
tff(fact_6914_mod__exE,axiom,
    ! [P: fun_li3352006551604695530i_assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(ex_ass463751140784270563_VEBTi(P)),H)
     => ~ ! [X3: list_VEBT_VEBTi] : ~ aa_Pro653082453696054616_nat_o(rep_assn(aa_lis2508312872377092907i_assn(P,X3)),H) ) ).

% mod_exE
tff(fact_6915_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(modulo_modulo_int(K),L)) ) ).

% neg_mod_bound
tff(fact_6916_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_6917_divmod__int__def,axiom,
    ! [M: num,Nb: num] : unique5052692396658037445od_int(M,Nb) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)),aa_int_int(modulo_modulo_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_int_def
tff(fact_6918_tanh__real__lt__1,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xc)),one_one_real) ).

% tanh_real_lt_1
tff(fact_6919_int__mod__ge,axiom,
    ! [A2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(modulo_modulo_int(A2),Nb)) ) ) ).

% int_mod_ge
tff(fact_6920_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(A2),B2)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(modulo_modulo_int(A2),B2)) ) ) ).

% neg_mod_conj
tff(fact_6921_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A2),B2))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A2),B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_6922_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(modulo_modulo_int(I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_6923_int__mod__eq,axiom,
    ! [B2: int,Nb: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),Nb)
       => ( ( aa_int_int(modulo_modulo_int(A2),Nb) = aa_int_int(modulo_modulo_int(B2),Nb) )
         => ( aa_int_int(modulo_modulo_int(A2),Nb) = B2 ) ) ) ) ).

% int_mod_eq
tff(fact_6924_int__mod__lem,axiom,
    ! [Nb: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),Nb) )
      <=> ( aa_int_int(modulo_modulo_int(B2),Nb) = B2 ) ) ) ).

% int_mod_lem
tff(fact_6925_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_6926_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_6927_int__mod__le_H,axiom,
    ! [B2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(minus_minus_int(B2),Nb))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(B2),Nb)),aa_int_int(minus_minus_int(B2),Nb)) ) ).

% int_mod_le'
tff(fact_6928_nonneg__mod__div,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A2),B2))
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ) ).

% nonneg_mod_div
tff(fact_6929_zdiv__mono__strict,axiom,
    ! [A: int,B: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => ( ( aa_int_int(modulo_modulo_int(A),Nb) = zero_zero_int )
         => ( ( aa_int_int(modulo_modulo_int(B),Nb) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_6930_div__mod__decomp__int,axiom,
    ! [A: int,Nb: int] : A = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),Nb)),Nb)),aa_int_int(modulo_modulo_int(A),Nb)) ).

% div_mod_decomp_int
tff(fact_6931_mod__div__equality__div__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(times_times_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),B2) = aa_int_int(minus_minus_int(A2),aa_int_int(modulo_modulo_int(A2),B2)) ).

% mod_div_equality_div_eq
tff(fact_6932_pos__mod__bound2,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% pos_mod_bound2
tff(fact_6933_int__mod__ge_H,axiom,
    ! [B2: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(B2),Nb)),aa_int_int(modulo_modulo_int(B2),Nb)) ) ) ).

% int_mod_ge'
tff(fact_6934_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(K),L)),zero_zero_int)
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(plus_plus_int(K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_6935_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(modulo_modulo_int(K),L) = aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_6936_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_6937_nat__mod__distrib,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => ( aa_int_nat(nat2,aa_int_int(modulo_modulo_int(Xc),Ya)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xc)),aa_int_nat(nat2,Ya)) ) ) ) ).

% nat_mod_distrib
tff(fact_6938_real__of__int__div__aux,axiom,
    ! [Xc: int,D2: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Xc)),aa_int_real(ring_1_of_int_real,D2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xc),D2))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(modulo_modulo_int(Xc),D2))),aa_int_real(ring_1_of_int_real,D2))) ).

% real_of_int_div_aux
tff(fact_6939_pos__mod__sign2,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% pos_mod_sign2
tff(fact_6940_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_6941_nmod2,axiom,
    ! [Nb: int] :
      ( ( aa_int_int(modulo_modulo_int(Nb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int )
      | ( aa_int_int(modulo_modulo_int(Nb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% nmod2
tff(fact_6942_mod__exp__less__eq__exp,axiom,
    ! [A2: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(modulo_modulo_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% mod_exp_less_eq_exp
tff(fact_6943_mod__power__lem,axiom,
    ! [A2: int,Nb: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A2)
     => ( aa_int_int(modulo_modulo_int(aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(A2),M)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb),zero_zero_int,aa_nat_int(power_power_int(A2),Nb)) ) ) ).

% mod_power_lem
tff(fact_6944_split__zmod,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(modulo_modulo_int(Nb),K))
    <=> ( ( ( K = zero_zero_int )
         => aa_int_o(P,Nb) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
         => ! [I2: int,J: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J),K)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J) ) )
             => aa_int_o(P,J) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: int,J: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),zero_zero_int)
                & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J) ) )
             => aa_int_o(P,J) ) ) ) ) ).

% split_zmod
tff(fact_6945_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),Q3)),R3) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R3),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R3)
         => ( aa_int_int(modulo_modulo_int(A2),B2) = R3 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_6946_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),Q3)),R3) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R3),B2)
         => ( aa_int_int(modulo_modulo_int(A2),B2) = R3 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_6947_mod__add__if__z,axiom,
    ! [Xc: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(Xc),Ya)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(Xc),Ya)),Z),aa_int_int(plus_plus_int(Xc),Ya),aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(Xc),Ya)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_6948_mod__sub__if__z,axiom,
    ! [Xc: int,Z: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),Z)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Z)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z)
             => ( aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(Xc),Ya)),Z) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Xc),aa_int_int(minus_minus_int(Xc),Ya),aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(Xc),Ya)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_6949_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C2)
     => ( aa_int_int(modulo_modulo_int(A2),aa_int_int(times_times_int(B2),C2)) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(B2),aa_int_int(modulo_modulo_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),C2))),aa_int_int(modulo_modulo_int(A2),B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_6950_axxmod2,axiom,
    ! [Xc: int] :
      ( ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(one_one_int),Xc)),Xc)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int )
      & ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(zero_zero_int),Xc)),Xc)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% axxmod2
tff(fact_6951_z1pmod2,axiom,
    ! [B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% z1pmod2
tff(fact_6952_verit__le__mono__div__int,axiom,
    ! [A: int,B: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(plus_plus_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),Nb)),
                $ite(aa_int_int(modulo_modulo_int(B),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_6953_split__neg__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K)),aa_int_int(modulo_modulo_int(Nb),K))
      <=> ! [I2: int,J: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),zero_zero_int)
              & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J) ) ) ) ).

% split_neg_lemma
tff(fact_6954_split__pos__lemma,axiom,
    ! [K: int,P: fun_int_fun_int_o,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),K)),aa_int_int(modulo_modulo_int(Nb),K))
      <=> ! [I2: int,J: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J),K)
              & ( Nb = aa_int_int(plus_plus_int(aa_int_int(times_times_int(K),I2)),J) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J) ) ) ) ).

% split_pos_lemma
tff(fact_6955_p1mod22k,axiom,
    ! [B2: int,Nb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),one_one_int)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(B2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))),one_one_int) ).

% p1mod22k
tff(fact_6956_p1mod22k_H,axiom,
    ! [B2: int,Nb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(B2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))) ).

% p1mod22k'
tff(fact_6957_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(Q3,R3))
    <=> ( ( K = aa_int_int(plus_plus_int(aa_int_int(times_times_int(L),Q3)),R3) )
        & $ite(
            aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L),
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),R3)
            & aa_int_o(aa_int_fun_int_o(ord_less_int,R3),L) ),
            $ite(
              aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int),
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),R3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,R3),zero_zero_int) ),
              Q3 = zero_zero_int ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_6958_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(B2),A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_6959_eme1p,axiom,
    ! [Nb: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D2)
         => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),Nb)),D2) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(modulo_modulo_int(Nb),D2)) ) ) ) ) ).

% eme1p
tff(fact_6960_emep1,axiom,
    ! [Nb: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),D2)
         => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(Nb),one_one_int)),D2) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(Nb),D2)),one_one_int) ) ) ) ) ).

% emep1
tff(fact_6961_even__flip__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(M),A2))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6962_even__flip__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),A2))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)
        <=> ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_6963_sb__inc__lem,axiom,
    ! [A2: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(plus_plus_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem
tff(fact_6964_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo2: option4927543243414619207at_nat,Deg: nat,Tree_list2: list_VEBT_VEBT,Summarya: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
      aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo2,Deg,Tree_list2),Summarya)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi2,Degi2,Tree_array2),Summaryi2)) = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_o_assn(pure_assn,
                ( ( Mmoi2 = Mmo2 )
                & ( Degi2 = Deg ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summarya),Summaryi2))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_eq(Tree_list2),Tree_array2))) ).

% vebt_assn_raw.simps(2)
tff(fact_6965_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(B2),one_one_int)),A2))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_6966_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A2,B2),product_Pair_int_int(Q3,R3))
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),product_Pair_int_int(Q3,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R3)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_6967_ln__one__minus__pos__lower__bound,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xc))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_6968_triangle__def,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% triangle_def
tff(fact_6969_obtain__set__pred,axiom,
    ! [Z: nat,Xc: nat,A: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Xc)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(A),Z)
       => ( aa_set_nat_o(finite_finite_nat,A)
         => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(A),Xc),X_12) ) ) ) ).

% obtain_set_pred
tff(fact_6970_obtain__set__succ,axiom,
    ! [Xc: nat,Z: nat,A: set_nat,B: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Z)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A),Z)
       => ( aa_set_nat_o(finite_finite_nat,B)
         => ( ( A = B )
           => ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A),Xc),X_12) ) ) ) ) ).

% obtain_set_succ
tff(fact_6971_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),Nb)
     => aa_set_nat_o(finite_finite_nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_6972_succ__none__empty,axiom,
    ! [Xs: set_nat,A2: nat] :
      ( ~ ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),A2),X_12)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),X4) ) ) ) ).

% succ_none_empty
tff(fact_6973_pred__none__empty,axiom,
    ! [Xs: set_nat,A2: nat] :
      ( ~ ? [X_12: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_pred_in_set(Xs),A2),X_12)
     => ( aa_set_nat_o(finite_finite_nat,Xs)
       => ~ ? [X4: nat] :
              ( aa_set_nat_o(member_nat(X4),Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),A2) ) ) ) ).

% pred_none_empty
tff(fact_6974_neg__equal__iff__equal,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(uminus_uminus_real,B2) )
    <=> ( A2 = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_6975_neg__equal__iff__equal,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(uminus_uminus_int,B2) )
    <=> ( A2 = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_6976_neg__equal__iff__equal,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(uminus1482373934393186551omplex,B2) )
    <=> ( A2 = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_6977_neg__equal__iff__equal,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger(A2) = uminus1351360451143612070nteger(B2) )
    <=> ( A2 = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_6978_neg__equal__iff__equal,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(uminus_uminus_rat,B2) )
    <=> ( A2 = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_6979_add_Oinverse__inverse,axiom,
    ! [A2: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,A2)) = A2 ).

% add.inverse_inverse
tff(fact_6980_add_Oinverse__inverse,axiom,
    ! [A2: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A2)) = A2 ).

% add.inverse_inverse
tff(fact_6981_add_Oinverse__inverse,axiom,
    ! [A2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A2)) = A2 ).

% add.inverse_inverse
tff(fact_6982_add_Oinverse__inverse,axiom,
    ! [A2: code_integer] : uminus1351360451143612070nteger(uminus1351360451143612070nteger(A2)) = A2 ).

% add.inverse_inverse
tff(fact_6983_add_Oinverse__inverse,axiom,
    ! [A2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,A2)) = A2 ).

% add.inverse_inverse
tff(fact_6984_neg__equal__zero,axiom,
    ! [A2: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = A2 )
    <=> ( A2 = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_6985_neg__equal__zero,axiom,
    ! [A2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = A2 )
    <=> ( A2 = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_6986_neg__equal__zero,axiom,
    ! [A2: code_integer] :
      ( ( uminus1351360451143612070nteger(A2) = A2 )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_6987_neg__equal__zero,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A2) = A2 )
    <=> ( A2 = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_6988_equal__neg__zero,axiom,
    ! [A2: real] :
      ( ( A2 = aa_real_real(uminus_uminus_real,A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_6989_equal__neg__zero,axiom,
    ! [A2: int] :
      ( ( A2 = aa_int_int(uminus_uminus_int,A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_6990_equal__neg__zero,axiom,
    ! [A2: code_integer] :
      ( ( A2 = uminus1351360451143612070nteger(A2) )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_6991_equal__neg__zero,axiom,
    ! [A2: rat] :
      ( ( A2 = aa_rat_rat(uminus_uminus_rat,A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_6992_neg__equal__0__iff__equal,axiom,
    ! [A2: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
tff(fact_6993_neg__equal__0__iff__equal,axiom,
    ! [A2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_6994_neg__equal__0__iff__equal,axiom,
    ! [A2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A2) = zero_zero_complex )
    <=> ( A2 = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
tff(fact_6995_neg__equal__0__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( uminus1351360451143612070nteger(A2) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_6996_neg__equal__0__iff__equal,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_6997_neg__0__equal__iff__equal,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(uminus_uminus_real,A2) )
    <=> ( zero_zero_real = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_6998_neg__0__equal__iff__equal,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A2) )
    <=> ( zero_zero_int = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_6999_neg__0__equal__iff__equal,axiom,
    ! [A2: complex] :
      ( ( zero_zero_complex = aa_complex_complex(uminus1482373934393186551omplex,A2) )
    <=> ( zero_zero_complex = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_7000_neg__0__equal__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger = uminus1351360451143612070nteger(A2) )
    <=> ( zero_z3403309356797280102nteger = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_7001_neg__0__equal__iff__equal,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = aa_rat_rat(uminus_uminus_rat,A2) )
    <=> ( zero_zero_rat = A2 ) ) ).

% neg_0_equal_iff_equal
tff(fact_7002_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_7003_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_7004_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_7005_add_Oinverse__neutral,axiom,
    uminus1351360451143612070nteger(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_7006_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_7007_neg__le__iff__le,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% neg_le_iff_le
tff(fact_7008_neg__le__iff__le,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B2)),uminus1351360451143612070nteger(A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2) ) ).

% neg_le_iff_le
tff(fact_7009_neg__le__iff__le,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% neg_le_iff_le
tff(fact_7010_neg__le__iff__le,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% neg_le_iff_le
tff(fact_7011_neg__numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( M = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_7012_neg__numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( M = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_7013_neg__numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( M = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_7014_neg__numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( uminus1351360451143612070nteger(numera6620942414471956472nteger(M)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) )
    <=> ( M = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_7015_neg__numeral__eq__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( M = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_7016_neg__less__iff__less,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% neg_less_iff_less
tff(fact_7017_neg__less__iff__less,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2) ) ).

% neg_less_iff_less
tff(fact_7018_neg__less__iff__less,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B2)),uminus1351360451143612070nteger(A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2) ) ).

% neg_less_iff_less
tff(fact_7019_neg__less__iff__less,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% neg_less_iff_less
tff(fact_7020_minus__add__distrib,axiom,
    ! [A2: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B2)) ).

% minus_add_distrib
tff(fact_7021_minus__add__distrib,axiom,
    ! [A2: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A2),B2)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) ).

% minus_add_distrib
tff(fact_7022_minus__add__distrib,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A2),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% minus_add_distrib
tff(fact_7023_minus__add__distrib,axiom,
    ! [A2: code_integer,B2: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A2,B2)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A2),uminus1351360451143612070nteger(B2)) ).

% minus_add_distrib
tff(fact_7024_minus__add__distrib,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A2),B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(uminus_uminus_rat,B2)) ).

% minus_add_distrib
tff(fact_7025_minus__add__cancel,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) = B2 ).

% minus_add_cancel
tff(fact_7026_minus__add__cancel,axiom,
    ! [A2: int,B2: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),aa_int_int(plus_plus_int(A2),B2)) = B2 ).

% minus_add_cancel
tff(fact_7027_minus__add__cancel,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(plus_plus_complex(A2),B2)) = B2 ).

% minus_add_cancel
tff(fact_7028_minus__add__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A2),plus_p5714425477246183910nteger(A2,B2)) = B2 ).

% minus_add_cancel
tff(fact_7029_minus__add__cancel,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(plus_plus_rat(A2),B2)) = B2 ).

% minus_add_cancel
tff(fact_7030_add__minus__cancel,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),B2)) = B2 ).

% add_minus_cancel
tff(fact_7031_add__minus__cancel,axiom,
    ! [A2: int,B2: int] : aa_int_int(plus_plus_int(A2),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),B2)) = B2 ).

% add_minus_cancel
tff(fact_7032_add__minus__cancel,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),B2)) = B2 ).

% add_minus_cancel
tff(fact_7033_add__minus__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(A2,plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A2),B2)) = B2 ).

% add_minus_cancel
tff(fact_7034_add__minus__cancel,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A2)),B2)) = B2 ).

% add_minus_cancel
tff(fact_7035_mult__minus__right,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) ).

% mult_minus_right
tff(fact_7036_mult__minus__right,axiom,
    ! [A2: int,B2: int] : aa_int_int(times_times_int(A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(times_times_int(A2),B2)) ).

% mult_minus_right
tff(fact_7037_mult__minus__right,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(times_times_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A2),B2)) ).

% mult_minus_right
tff(fact_7038_mult__minus__right,axiom,
    ! [A2: code_integer,B2: code_integer] : times_3573771949741848930nteger(A2,uminus1351360451143612070nteger(B2)) = uminus1351360451143612070nteger(times_3573771949741848930nteger(A2,B2)) ).

% mult_minus_right
tff(fact_7039_mult__minus__right,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(A2),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A2),B2)) ).

% mult_minus_right
tff(fact_7040_minus__mult__minus,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2) ).

% minus_mult_minus
tff(fact_7041_minus__mult__minus,axiom,
    ! [A2: int,B2: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(times_times_int(A2),B2) ).

% minus_mult_minus
tff(fact_7042_minus__mult__minus,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(times_times_complex(A2),B2) ).

% minus_mult_minus
tff(fact_7043_minus__mult__minus,axiom,
    ! [A2: code_integer,B2: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(A2),uminus1351360451143612070nteger(B2)) = times_3573771949741848930nteger(A2,B2) ).

% minus_mult_minus
tff(fact_7044_minus__mult__minus,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(times_times_rat(A2),B2) ).

% minus_mult_minus
tff(fact_7045_mult__minus__left,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A2)),B2) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) ).

% mult_minus_left
tff(fact_7046_mult__minus__left,axiom,
    ! [A2: int,B2: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(uminus_uminus_int,aa_int_int(times_times_int(A2),B2)) ).

% mult_minus_left
tff(fact_7047_mult__minus__left,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),B2) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A2),B2)) ).

% mult_minus_left
tff(fact_7048_mult__minus__left,axiom,
    ! [A2: code_integer,B2: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(A2),B2) = uminus1351360451143612070nteger(times_3573771949741848930nteger(A2,B2)) ).

% mult_minus_left
tff(fact_7049_mult__minus__left,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A2)),B2) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(times_times_rat(A2),B2)) ).

% mult_minus_left
tff(fact_7050_minus__diff__eq,axiom,
    ! [A2: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2) ).

% minus_diff_eq
tff(fact_7051_minus__diff__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(minus_minus_int(A2),B2)) = aa_int_int(minus_minus_int(B2),A2) ).

% minus_diff_eq
tff(fact_7052_minus__diff__eq,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A2),B2)) = aa_complex_complex(minus_minus_complex(B2),A2) ).

% minus_diff_eq
tff(fact_7053_minus__diff__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : uminus1351360451143612070nteger(minus_8373710615458151222nteger(A2,B2)) = minus_8373710615458151222nteger(B2,A2) ).

% minus_diff_eq
tff(fact_7054_minus__diff__eq,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A2),B2)) = aa_rat_rat(minus_minus_rat(B2),A2) ).

% minus_diff_eq
tff(fact_7055_div__minus__minus,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) ).

% div_minus_minus
tff(fact_7056_div__minus__minus,axiom,
    ! [A2: code_integer,B2: code_integer] : divide6298287555418463151nteger(uminus1351360451143612070nteger(A2),uminus1351360451143612070nteger(B2)) = divide6298287555418463151nteger(A2,B2) ).

% div_minus_minus
tff(fact_7057_minus__dvd__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(uminus_uminus_real,Xc)),Ya)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xc),Ya) ) ).

% minus_dvd_iff
tff(fact_7058_minus__dvd__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,Xc)),Ya)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xc),Ya) ) ).

% minus_dvd_iff
tff(fact_7059_minus__dvd__iff,axiom,
    ! [Xc: complex,Ya: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(uminus1482373934393186551omplex,Xc)),Ya)
    <=> aa_complex_o(dvd_dvd_complex(Xc),Ya) ) ).

% minus_dvd_iff
tff(fact_7060_minus__dvd__iff,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,uminus1351360451143612070nteger(Xc)),Ya)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xc),Ya) ) ).

% minus_dvd_iff
tff(fact_7061_minus__dvd__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,Xc)),Ya)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xc),Ya) ) ).

% minus_dvd_iff
tff(fact_7062_dvd__minus__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xc),aa_real_real(uminus_uminus_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xc),Ya) ) ).

% dvd_minus_iff
tff(fact_7063_dvd__minus__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xc),aa_int_int(uminus_uminus_int,Ya))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xc),Ya) ) ).

% dvd_minus_iff
tff(fact_7064_dvd__minus__iff,axiom,
    ! [Xc: complex,Ya: complex] :
      ( aa_complex_o(dvd_dvd_complex(Xc),aa_complex_complex(uminus1482373934393186551omplex,Ya))
    <=> aa_complex_o(dvd_dvd_complex(Xc),Ya) ) ).

% dvd_minus_iff
tff(fact_7065_dvd__minus__iff,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xc),uminus1351360451143612070nteger(Ya))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xc),Ya) ) ).

% dvd_minus_iff
tff(fact_7066_dvd__minus__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xc),aa_rat_rat(uminus_uminus_rat,Ya))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xc),Ya) ) ).

% dvd_minus_iff
tff(fact_7067_mod__minus__minus,axiom,
    ! [A2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A2),B2)) ).

% mod_minus_minus
tff(fact_7068_mod__minus__minus,axiom,
    ! [A2: code_integer,B2: code_integer] : modulo364778990260209775nteger(uminus1351360451143612070nteger(A2),uminus1351360451143612070nteger(B2)) = uminus1351360451143612070nteger(modulo364778990260209775nteger(A2,B2)) ).

% mod_minus_minus
tff(fact_7069_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,set_VEBT_VEBT2(Xs)) ).

% List.finite_set
tff(fact_7070_List_Ofinite__set,axiom,
    ! [Xs: list_real] : aa_set_real_o(finite_finite_real,set_real2(Xs)) ).

% List.finite_set
tff(fact_7071_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : aa_set_nat_o(finite_finite_nat,set_nat2(Xs)) ).

% List.finite_set
tff(fact_7072_List_Ofinite__set,axiom,
    ! [Xs: list_int] : aa_set_int_o(finite_finite_int,set_int2(Xs)) ).

% List.finite_set
tff(fact_7073_List_Ofinite__set,axiom,
    ! [Xs: list_Code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_Code_integer2(Xs)) ).

% List.finite_set
tff(fact_7074_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : aa_set_complex_o(finite3207457112153483333omplex,set_complex2(Xs)) ).

% List.finite_set
tff(fact_7075_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) ).

% finite_atLeastLessThan
tff(fact_7076_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) ).

% finite_atLeastAtMost
tff(fact_7077_triangle__0,axiom,
    aa_nat_nat(nat_triangle,zero_zero_nat) = zero_zero_nat ).

% triangle_0
tff(fact_7078_neg__0__le__iff__le,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% neg_0_le_iff_le
tff(fact_7079_neg__0__le__iff__le,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_7080_neg__0__le__iff__le,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_7081_neg__0__le__iff__le,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_7082_neg__le__0__iff__le,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% neg_le_0_iff_le
tff(fact_7083_neg__le__0__iff__le,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A2)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ).

% neg_le_0_iff_le
tff(fact_7084_neg__le__0__iff__le,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% neg_le_0_iff_le
tff(fact_7085_neg__le__0__iff__le,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% neg_le_0_iff_le
tff(fact_7086_less__eq__neg__nonpos,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) ) ).

% less_eq_neg_nonpos
tff(fact_7087_less__eq__neg__nonpos,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),uminus1351360451143612070nteger(A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_7088_less__eq__neg__nonpos,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_7089_less__eq__neg__nonpos,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_7090_neg__less__eq__nonneg,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% neg_less_eq_nonneg
tff(fact_7091_neg__less__eq__nonneg,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A2)),A2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ).

% neg_less_eq_nonneg
tff(fact_7092_neg__less__eq__nonneg,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% neg_less_eq_nonneg
tff(fact_7093_neg__less__eq__nonneg,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% neg_less_eq_nonneg
tff(fact_7094_neg__less__0__iff__less,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A2)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ).

% neg_less_0_iff_less
tff(fact_7095_neg__less__0__iff__less,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A2)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ).

% neg_less_0_iff_less
tff(fact_7096_neg__less__0__iff__less,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A2)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2) ) ).

% neg_less_0_iff_less
tff(fact_7097_neg__less__0__iff__less,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% neg_less_0_iff_less
tff(fact_7098_neg__0__less__iff__less,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% neg_0_less_iff_less
tff(fact_7099_neg__0__less__iff__less,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_7100_neg__0__less__iff__less,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_7101_neg__0__less__iff__less,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_7102_neg__less__pos,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A2)),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2) ) ).

% neg_less_pos
tff(fact_7103_neg__less__pos,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A2)),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2) ) ).

% neg_less_pos
tff(fact_7104_neg__less__pos,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A2)),A2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2) ) ).

% neg_less_pos
tff(fact_7105_neg__less__pos,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2) ) ).

% neg_less_pos
tff(fact_7106_less__neg__neg,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(uminus_uminus_real,A2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real) ) ).

% less_neg_neg
tff(fact_7107_less__neg__neg,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int) ) ).

% less_neg_neg
tff(fact_7108_less__neg__neg,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),uminus1351360451143612070nteger(A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_7109_less__neg__neg,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(uminus_uminus_rat,A2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_7110_ab__left__minus,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),A2) = zero_zero_real ).

% ab_left_minus
tff(fact_7111_ab__left__minus,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),A2) = zero_zero_int ).

% ab_left_minus
tff(fact_7112_ab__left__minus,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),A2) = zero_zero_complex ).

% ab_left_minus
tff(fact_7113_ab__left__minus,axiom,
    ! [A2: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A2),A2) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_7114_ab__left__minus,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A2)),A2) = zero_zero_rat ).

% ab_left_minus
tff(fact_7115_add_Oright__inverse,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(uminus_uminus_real,A2)) = zero_zero_real ).

% add.right_inverse
tff(fact_7116_add_Oright__inverse,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(A2),aa_int_int(uminus_uminus_int,A2)) = zero_zero_int ).

% add.right_inverse
tff(fact_7117_add_Oright__inverse,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,A2)) = zero_zero_complex ).

% add.right_inverse
tff(fact_7118_add_Oright__inverse,axiom,
    ! [A2: code_integer] : plus_p5714425477246183910nteger(A2,uminus1351360451143612070nteger(A2)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_7119_add_Oright__inverse,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(uminus_uminus_rat,A2)) = zero_zero_rat ).

% add.right_inverse
tff(fact_7120_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_7121_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_7122_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(M)),numera6690914467698888265omplex(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_7123_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = uminus1351360451143612070nteger(plus_p5714425477246183910nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_7124_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_num_rat(numeral_numeral_rat,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_7125_diff__0,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),A2) = aa_real_real(uminus_uminus_real,A2) ).

% diff_0
tff(fact_7126_diff__0,axiom,
    ! [A2: int] : aa_int_int(minus_minus_int(zero_zero_int),A2) = aa_int_int(uminus_uminus_int,A2) ).

% diff_0
tff(fact_7127_diff__0,axiom,
    ! [A2: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A2) = aa_complex_complex(uminus1482373934393186551omplex,A2) ).

% diff_0
tff(fact_7128_diff__0,axiom,
    ! [A2: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A2) = uminus1351360451143612070nteger(A2) ).

% diff_0
tff(fact_7129_diff__0,axiom,
    ! [A2: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),A2) = aa_rat_rat(uminus_uminus_rat,A2) ).

% diff_0
tff(fact_7130_verit__minus__simplify_I3_J,axiom,
    ! [B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B2) = aa_real_real(uminus_uminus_real,B2) ).

% verit_minus_simplify(3)
tff(fact_7131_verit__minus__simplify_I3_J,axiom,
    ! [B2: int] : aa_int_int(minus_minus_int(zero_zero_int),B2) = aa_int_int(uminus_uminus_int,B2) ).

% verit_minus_simplify(3)
tff(fact_7132_verit__minus__simplify_I3_J,axiom,
    ! [B2: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B2) = aa_complex_complex(uminus1482373934393186551omplex,B2) ).

% verit_minus_simplify(3)
tff(fact_7133_verit__minus__simplify_I3_J,axiom,
    ! [B2: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B2) = uminus1351360451143612070nteger(B2) ).

% verit_minus_simplify(3)
tff(fact_7134_verit__minus__simplify_I3_J,axiom,
    ! [B2: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B2) = aa_rat_rat(uminus_uminus_rat,B2) ).

% verit_minus_simplify(3)
tff(fact_7135_mult__minus1,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,one_one_real)),Z) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1
tff(fact_7136_mult__minus1,axiom,
    ! [Z: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,one_one_int)),Z) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1
tff(fact_7137_mult__minus1,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Z) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1
tff(fact_7138_mult__minus1,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(one_one_Code_integer),Z) = uminus1351360451143612070nteger(Z) ).

% mult_minus1
tff(fact_7139_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Z) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1
tff(fact_7140_mult__minus1__right,axiom,
    ! [Z: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Z) ).

% mult_minus1_right
tff(fact_7141_mult__minus1__right,axiom,
    ! [Z: int] : aa_int_int(times_times_int(Z),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Z) ).

% mult_minus1_right
tff(fact_7142_mult__minus1__right,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Z) ).

% mult_minus1_right
tff(fact_7143_mult__minus1__right,axiom,
    ! [Z: code_integer] : times_3573771949741848930nteger(Z,uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(Z) ).

% mult_minus1_right
tff(fact_7144_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(times_times_rat(Z),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Z) ).

% mult_minus1_right
tff(fact_7145_uminus__add__conv__diff,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2) ).

% uminus_add_conv_diff
tff(fact_7146_uminus__add__conv__diff,axiom,
    ! [A2: int,B2: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(minus_minus_int(B2),A2) ).

% uminus_add_conv_diff
tff(fact_7147_uminus__add__conv__diff,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),B2) = aa_complex_complex(minus_minus_complex(B2),A2) ).

% uminus_add_conv_diff
tff(fact_7148_uminus__add__conv__diff,axiom,
    ! [A2: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A2),B2) = minus_8373710615458151222nteger(B2,A2) ).

% uminus_add_conv_diff
tff(fact_7149_uminus__add__conv__diff,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A2)),B2) = aa_rat_rat(minus_minus_rat(B2),A2) ).

% uminus_add_conv_diff
tff(fact_7150_diff__minus__eq__add,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) ).

% diff_minus_eq_add
tff(fact_7151_diff__minus__eq__add,axiom,
    ! [A2: int,B2: int] : aa_int_int(minus_minus_int(A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(plus_plus_int(A2),B2) ).

% diff_minus_eq_add
tff(fact_7152_diff__minus__eq__add,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(minus_minus_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(plus_plus_complex(A2),B2) ).

% diff_minus_eq_add
tff(fact_7153_diff__minus__eq__add,axiom,
    ! [A2: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A2,uminus1351360451143612070nteger(B2)) = plus_p5714425477246183910nteger(A2,B2) ).

% diff_minus_eq_add
tff(fact_7154_diff__minus__eq__add,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(plus_plus_rat(A2),B2) ).

% diff_minus_eq_add
tff(fact_7155_divide__minus1,axiom,
    ! [Xc: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Xc) ).

% divide_minus1
tff(fact_7156_divide__minus1,axiom,
    ! [Xc: complex] : aa_complex_complex(divide1717551699836669952omplex(Xc),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Xc) ).

% divide_minus1
tff(fact_7157_divide__minus1,axiom,
    ! [Xc: rat] : aa_rat_rat(divide_divide_rat(Xc),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Xc) ).

% divide_minus1
tff(fact_7158_div__minus1__right,axiom,
    ! [A2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A2) ).

% div_minus1_right
tff(fact_7159_div__minus1__right,axiom,
    ! [A2: code_integer] : divide6298287555418463151nteger(A2,uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(A2) ).

% div_minus1_right
tff(fact_7160_minus__mod__self1,axiom,
    ! [B2: int,A2: int] : aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(B2),A2)),B2) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A2)),B2) ).

% minus_mod_self1
tff(fact_7161_minus__mod__self1,axiom,
    ! [B2: code_integer,A2: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B2,A2),B2) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A2),B2) ).

% minus_mod_self1
tff(fact_7162_infinite__Icc__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A2,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% infinite_Icc_iff
tff(fact_7163_infinite__Icc__iff,axiom,
    ! [A2: real,B2: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A2,B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% infinite_Icc_iff
tff(fact_7164_infinite__Ico__iff,axiom,
    ! [A2: real,B2: real] :
      ( ~ aa_set_real_o(finite_finite_real,set_or66887138388493659n_real(A2,B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ).

% infinite_Ico_iff
tff(fact_7165_infinite__Ico__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ aa_set_rat_o(finite_finite_rat,set_or4029947393144176647an_rat(A2,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2) ) ).

% infinite_Ico_iff
tff(fact_7166_real__add__minus__iff,axiom,
    ! [Xc: real,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(uminus_uminus_real,A2)) = zero_zero_real )
    <=> ( Xc = A2 ) ) ).

% real_add_minus_iff
tff(fact_7167_triangle__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(nat_triangle,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,Nb)),aa_nat_nat(suc,Nb)) ).

% triangle_Suc
tff(fact_7168_listI__assn__finite,axiom,
    ! [I3: set_nat,A: fun_VE905575822219544865i_assn,Xs: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] :
      ( ~ aa_set_nat_o(finite_finite_nat,I3)
     => ( aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(I3,A,Xs),Xsi) = bot_bot_assn ) ) ).

% listI_assn_finite
tff(fact_7169_add__neg__numeral__special_I7_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% add_neg_numeral_special(7)
tff(fact_7170_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(plus_plus_int(one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% add_neg_numeral_special(7)
tff(fact_7171_add__neg__numeral__special_I7_J,axiom,
    aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% add_neg_numeral_special(7)
tff(fact_7172_add__neg__numeral__special_I7_J,axiom,
    plus_p5714425477246183910nteger(one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_7173_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_7174_add__neg__numeral__special_I8_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = zero_zero_real ).

% add_neg_numeral_special(8)
tff(fact_7175_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).

% add_neg_numeral_special(8)
tff(fact_7176_add__neg__numeral__special_I8_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = zero_zero_complex ).

% add_neg_numeral_special(8)
tff(fact_7177_add__neg__numeral__special_I8_J,axiom,
    plus_p5714425477246183910nteger(uminus1351360451143612070nteger(one_one_Code_integer),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_7178_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_7179_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7180_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7181_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7182_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) = uminus1351360451143612070nteger(one_one_Code_integer) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7183_numeral__eq__neg__one__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( Nb = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_7184_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7185_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7186_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7187_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( uminus1351360451143612070nteger(one_one_Code_integer) = uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7188_neg__one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Nb = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_7189_diff__numeral__special_I12_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = zero_zero_real ).

% diff_numeral_special(12)
tff(fact_7190_diff__numeral__special_I12_J,axiom,
    aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_7191_diff__numeral__special_I12_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = zero_zero_complex ).

% diff_numeral_special(12)
tff(fact_7192_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(uminus1351360451143612070nteger(one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_7193_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_7194_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_7195_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A2: int] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_7196_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A2: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_7197_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A2: code_integer] : times_3573771949741848930nteger(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),Nb),times_3573771949741848930nteger(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),Nb),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_7198_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A2: rat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),A2)) = A2 ).

% left_minus_one_mult_self
tff(fact_7199_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)) = one_one_real ).

% minus_one_mult_self
tff(fact_7200_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)) = one_one_int ).

% minus_one_mult_self
tff(fact_7201_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)) = one_one_complex ).

% minus_one_mult_self
tff(fact_7202_minus__one__mult__self,axiom,
    ! [Nb: nat] : times_3573771949741848930nteger(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),Nb),power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),Nb)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_7203_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)) = one_one_rat ).

% minus_one_mult_self
tff(fact_7204_mod__minus1__right,axiom,
    ! [A2: int] : aa_int_int(modulo_modulo_int(A2),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_7205_mod__minus1__right,axiom,
    ! [A2: code_integer] : modulo364778990260209775nteger(A2,uminus1351360451143612070nteger(one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_7206_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(4)
tff(fact_7207_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(U)),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(U))),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))),uminus1351360451143612070nteger(numera6620942414471956472nteger(V)),uminus1351360451143612070nteger(numera6620942414471956472nteger(U))) ).

% max_number_of(4)
tff(fact_7208_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(4)
tff(fact_7209_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(4)
tff(fact_7210_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))),aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,V),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,U))) ).

% max_number_of(3)
tff(fact_7211_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(U)),numera6620942414471956472nteger(V)) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(U))),numera6620942414471956472nteger(V)),numera6620942414471956472nteger(V),uminus1351360451143612070nteger(numera6620942414471956472nteger(U))) ).

% max_number_of(3)
tff(fact_7212_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))),aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,V),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,U))) ).

% max_number_of(3)
tff(fact_7213_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))),aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,U))) ).

% max_number_of(3)
tff(fact_7214_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_real_real(aa_rea7340605660300375871l_real(ord_max_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,U)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V)),aa_num_real(numeral_numeral_real,U)) ).

% max_number_of(2)
tff(fact_7215_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ord_max_Code_integer(numera6620942414471956472nteger(U),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(U)),uminus1351360451143612070nteger(numera6620942414471956472nteger(V))),uminus1351360451143612070nteger(numera6620942414471956472nteger(V)),numera6620942414471956472nteger(U)) ).

% max_number_of(2)
tff(fact_7216_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_rat_rat(ord_max_rat(aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,U)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_num_rat(numeral_numeral_rat,U)) ).

% max_number_of(2)
tff(fact_7217_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      aa_int_int(ord_max_int(aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,U)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,U)) ).

% max_number_of(2)
tff(fact_7218_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_7219_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_7220_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W)))),Ya) ).

% semiring_norm(168)
tff(fact_7221_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Ya: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Ya)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W)))),Ya) ).

% semiring_norm(168)
tff(fact_7222_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Ya: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Ya)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W)))),Ya) ).

% semiring_norm(168)
tff(fact_7223_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Ya: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(V)),plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(W)),Ya)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(plus_plus_num(V),W))),Ya) ).

% semiring_norm(168)
tff(fact_7224_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Ya: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Ya)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W)))),Ya) ).

% semiring_norm(168)
tff(fact_7225_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_7226_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_7227_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),Nb)) ).

% diff_numeral_simps(2)
tff(fact_7228_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),Nb)) ).

% diff_numeral_simps(2)
tff(fact_7229_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),Nb)) ).

% diff_numeral_simps(2)
tff(fact_7230_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(M),Nb)) ).

% diff_numeral_simps(2)
tff(fact_7231_diff__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),Nb)) ).

% diff_numeral_simps(2)
tff(fact_7232_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),Nb))) ).

% diff_numeral_simps(3)
tff(fact_7233_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),Nb))) ).

% diff_numeral_simps(3)
tff(fact_7234_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),Nb))) ).

% diff_numeral_simps(3)
tff(fact_7235_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Nb)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(plus_plus_num(M),Nb))) ).

% diff_numeral_simps(3)
tff(fact_7236_diff__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),Nb))) ).

% diff_numeral_simps(3)
tff(fact_7237_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_7238_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_7239_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(times_times_num(M),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_7240_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Nb: num] : times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(times_times_num(M),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_7241_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_7242_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_7243_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_7244_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(M),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_7245_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),numera6620942414471956472nteger(Nb)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(times_times_num(M),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_7246_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_7247_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(M),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_7248_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(M),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_7249_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(M)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(M),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_7250_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : times_3573771949741848930nteger(numera6620942414471956472nteger(M),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(times_times_num(M),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_7251_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(M),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_7252_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W)))),Ya) ).

% semiring_norm(170)
tff(fact_7253_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Ya: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),Ya)) = aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W)))),Ya) ).

% semiring_norm(170)
tff(fact_7254_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Ya: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Ya)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(V),W)))),Ya) ).

% semiring_norm(170)
tff(fact_7255_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Ya: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(V)),times_3573771949741848930nteger(numera6620942414471956472nteger(W),Ya)) = times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(times_times_num(V),W))),Ya) ).

% semiring_norm(170)
tff(fact_7256_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Ya: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),Ya)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W)))),Ya) ).

% semiring_norm(170)
tff(fact_7257_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W)))),Ya) ).

% semiring_norm(171)
tff(fact_7258_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Ya: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,V)),aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Ya)) = aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W)))),Ya) ).

% semiring_norm(171)
tff(fact_7259_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Ya: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Ya)) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(V),W)))),Ya) ).

% semiring_norm(171)
tff(fact_7260_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Ya: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(V),times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(W)),Ya)) = times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(times_times_num(V),W))),Ya) ).

% semiring_norm(171)
tff(fact_7261_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Ya: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Ya)) = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W)))),Ya) ).

% semiring_norm(171)
tff(fact_7262_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W))),Ya) ).

% semiring_norm(172)
tff(fact_7263_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Ya: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Ya)) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W))),Ya) ).

% semiring_norm(172)
tff(fact_7264_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Ya: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),Ya)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W))),Ya) ).

% semiring_norm(172)
tff(fact_7265_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Ya: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(V)),times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(W)),Ya)) = times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(times_times_num(V),W)),Ya) ).

% semiring_norm(172)
tff(fact_7266_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Ya: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Ya)) = aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Ya) ).

% semiring_norm(172)
tff(fact_7267_neg__numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),M) ) ).

% neg_numeral_le_iff
tff(fact_7268_neg__numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),M) ) ).

% neg_numeral_le_iff
tff(fact_7269_neg__numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),M) ) ).

% neg_numeral_le_iff
tff(fact_7270_neg__numeral__le__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),M) ) ).

% neg_numeral_le_iff
tff(fact_7271_neg__numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),M) ) ).

% neg_numeral_less_iff
tff(fact_7272_neg__numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),M) ) ).

% neg_numeral_less_iff
tff(fact_7273_neg__numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),M) ) ).

% neg_numeral_less_iff
tff(fact_7274_neg__numeral__less__iff,axiom,
    ! [M: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),M) ) ).

% neg_numeral_less_iff
tff(fact_7275_round__neg__numeral,axiom,
    ! [Nb: num] : aa_real_int(archim8280529875227126926d_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_7276_round__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_int(archim7778729529865785530nd_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% round_neg_numeral
tff(fact_7277_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7278_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),uminus1351360451143612070nteger(numera6620942414471956472nteger(M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7279_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7280_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)))
    <=> ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_7281_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7282_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7283_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(one_one_Code_integer))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7284_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_7285_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( A2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = B2,A2 = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_7286_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: complex,B2: complex,W: num] :
      ( ( A2 = aa_complex_complex(divide1717551699836669952omplex(B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,aa_complex_complex(times_times_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = B2,A2 = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_7287_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( A2 = aa_rat_rat(divide_divide_rat(B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,aa_rat_rat(times_times_rat(A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B2,A2 = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_7288_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = A2 )
    <=> $ite(aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) != zero_zero_real,B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),A2 = zero_zero_real) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_7289_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: complex,W: num,A2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A2 )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),A2 = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_7290_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = A2 )
    <=> $ite(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) != zero_zero_rat,B2 = aa_rat_rat(times_times_rat(A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),A2 = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_7291_le__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_7292_le__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(divide_divide_rat(B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_7293_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B2) ) ).

% divide_le_eq_numeral1(2)
tff(fact_7294_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B2) ) ).

% divide_le_eq_numeral1(2)
tff(fact_7295_less__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_7296_less__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(divide_divide_rat(B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_7297_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))),B2) ) ).

% divide_less_eq_numeral1(2)
tff(fact_7298_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B2) ) ).

% divide_less_eq_numeral1(2)
tff(fact_7299_power2__minus,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_7300_power2__minus,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_7301_power2__minus,axiom,
    ! [A2: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_7302_power2__minus,axiom,
    ! [A2: code_integer] : power_8256067586552552935nteger(uminus1351360451143612070nteger(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_7303_power2__minus,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_7304_add__neg__numeral__special_I9_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_7305_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_7306_add__neg__numeral__special_I9_J,axiom,
    aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_7307_add__neg__numeral__special_I9_J,axiom,
    plus_p5714425477246183910nteger(uminus1351360451143612070nteger(one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_7308_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_7309_diff__numeral__special_I10_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_7310_diff__numeral__special_I10_J,axiom,
    aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_7311_diff__numeral__special_I10_J,axiom,
    aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_7312_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(uminus1351360451143612070nteger(one_one_Code_integer),one_one_Code_integer) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_7313_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_7314_diff__numeral__special_I11_J,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_7315_diff__numeral__special_I11_J,axiom,
    aa_int_int(minus_minus_int(one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_7316_diff__numeral__special_I11_J,axiom,
    aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_7317_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_7318_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_7319_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_7320_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(uminus1351360451143612070nteger(one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_7321_minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_7322_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(uminus1351360451143612070nteger(one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_7323_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_7324_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(uminus1351360451143612070nteger(one_one_Code_integer),numera6620942414471956472nteger(aa_num_num(bit0,one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_7325_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_7326_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_7327_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_complex(power_power_complex(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_7328_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: code_integer,Nb: nat] : power_8256067586552552935nteger(uminus1351360451143612070nteger(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = power_8256067586552552935nteger(A2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_7329_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% Power.ring_1_class.power_minus_even
tff(fact_7330_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_7331_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),Nb) = aa_nat_int(power_power_int(A2),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_7332_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),Nb) = aa_nat_complex(power_power_complex(A2),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_7333_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( power_8256067586552552935nteger(uminus1351360451143612070nteger(A2),Nb) = power_8256067586552552935nteger(A2,Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_7334_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [Nb: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A2)),Nb) = aa_nat_rat(power_power_rat(A2),Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_7335_power__minus__odd,axiom,
    ! [Nb: nat,A2: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),Nb) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ) ) ).

% power_minus_odd
tff(fact_7336_power__minus__odd,axiom,
    ! [Nb: nat,A2: int] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),Nb) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A2),Nb)) ) ) ).

% power_minus_odd
tff(fact_7337_power__minus__odd,axiom,
    ! [Nb: nat,A2: complex] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A2),Nb)) ) ) ).

% power_minus_odd
tff(fact_7338_power__minus__odd,axiom,
    ! [Nb: nat,A2: code_integer] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( power_8256067586552552935nteger(uminus1351360451143612070nteger(A2),Nb) = uminus1351360451143612070nteger(power_8256067586552552935nteger(A2,Nb)) ) ) ).

% power_minus_odd
tff(fact_7339_power__minus__odd,axiom,
    ! [Nb: nat,A2: rat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A2)),Nb) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A2),Nb)) ) ) ).

% power_minus_odd
tff(fact_7340_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_7341_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_int_int(minus_minus_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_7342_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_7343_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_7344_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_7345_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_7346_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_7347_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_7348_diff__numeral__special_I4_J,axiom,
    ! [M: num] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(M)),one_one_Code_integer) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_7349_diff__numeral__special_I4_J,axiom,
    ! [M: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(M),one))) ).

% diff_numeral_special(4)
tff(fact_7350_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Ya: int,Xc: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Ya) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xc))),Nb) )
    <=> ( Ya = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7351_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Ya: int,Xc: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Ya) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) )
    <=> ( Ya = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7352_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Ya: int,Xc: num,Nb: nat] :
      ( ( ring_17405671764205052669omplex(Ya) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xc))),Nb) )
    <=> ( Ya = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7353_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Ya: int,Xc: num,Nb: nat] :
      ( ( ring_18347121197199848620nteger(Ya) = power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xc)),Nb) )
    <=> ( Ya = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7354_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Ya: int,Xc: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Ya) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xc))),Nb) )
    <=> ( Ya = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_7355_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xc))),Nb) = aa_int_real(ring_1_of_int_real,Ya) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) = Ya ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7356_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) = aa_int_int(ring_1_of_int_int,Ya) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) = Ya ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7357_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xc))),Nb) = ring_17405671764205052669omplex(Ya) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) = Ya ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7358_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: int] :
      ( ( power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xc)),Nb) = ring_18347121197199848620nteger(Ya) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) = Ya ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7359_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,Ya: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xc))),Nb) = aa_int_rat(ring_1_of_int_rat,Ya) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb) = Ya ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_7360_ceiling__le__neg__numeral,axiom,
    ! [Xc: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xc)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_7361_ceiling__le__neg__numeral,axiom,
    ! [Xc: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_7362_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),Xc) ) ).

% neg_numeral_less_ceiling
tff(fact_7363_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xc))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),Xc) ) ).

% neg_numeral_less_ceiling
tff(fact_7364_ceiling__less__zero,axiom,
    ! [Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xc)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_7365_ceiling__less__zero,axiom,
    ! [Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_7366_zero__le__ceiling,axiom,
    ! [Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc) ) ).

% zero_le_ceiling
tff(fact_7367_zero__le__ceiling,axiom,
    ! [Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xc))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xc) ) ).

% zero_le_ceiling
tff(fact_7368_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_real ).

% power_minus1_even
tff(fact_7369_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_int ).

% power_minus1_even
tff(fact_7370_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_complex ).

% power_minus1_even
tff(fact_7371_power__minus1__even,axiom,
    ! [Nb: nat] : power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_7372_power__minus1__even,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = one_one_rat ).

% power_minus1_even
tff(fact_7373_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = one_one_real ) ) ).

% neg_one_even_power
tff(fact_7374_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = one_one_int ) ) ).

% neg_one_even_power
tff(fact_7375_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = one_one_complex ) ) ).

% neg_one_even_power
tff(fact_7376_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),Nb) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_7377_neg__one__even__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_7378_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% neg_one_odd_power
tff(fact_7379_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% neg_one_odd_power
tff(fact_7380_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% neg_one_odd_power
tff(fact_7381_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),Nb) = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_7382_neg__one__odd__power,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_7383_ceiling__less__neg__numeral,axiom,
    ! [Xc: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xc)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_7384_ceiling__less__neg__numeral,axiom,
    ! [Xc: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xc)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_7385_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xc: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),Xc) ) ).

% neg_numeral_le_ceiling
tff(fact_7386_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xc: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xc))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),Xc) ) ).

% neg_numeral_le_ceiling
tff(fact_7387_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xc))),Nb)),aa_int_real(ring_1_of_int_real,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)),A2) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7388_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xc)),Nb)),ring_18347121197199848620nteger(A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)),A2) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7389_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xc))),Nb)),aa_int_rat(ring_1_of_int_rat,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)),A2) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7390_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)),aa_int_int(ring_1_of_int_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)),A2) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_7391_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xc))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7392_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ring_18347121197199848620nteger(A2)),power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xc)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7393_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xc))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7394_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_7395_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xc))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7396_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A2)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7397_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ring_18347121197199848620nteger(A2)),power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xc)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7398_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xc: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A2)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xc))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_7399_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xc))),Nb)),aa_int_real(ring_1_of_int_real,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)),A2) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7400_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)),aa_int_int(ring_1_of_int_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)),A2) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7401_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(Xc)),Nb)),ring_18347121197199848620nteger(A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)),A2) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7402_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xc: num,Nb: nat,A2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xc))),Nb)),aa_int_rat(ring_1_of_int_rat,A2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xc))),Nb)),A2) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_7403_le__imp__neg__le,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2)) ) ).

% le_imp_neg_le
tff(fact_7404_le__imp__neg__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B2)),uminus1351360451143612070nteger(A2)) ) ).

% le_imp_neg_le
tff(fact_7405_le__imp__neg__le,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A2)) ) ).

% le_imp_neg_le
tff(fact_7406_le__imp__neg__le,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2)) ) ).

% le_imp_neg_le
tff(fact_7407_minus__le__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),A2) ) ).

% minus_le_iff
tff(fact_7408_minus__le__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A2)),B2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(B2)),A2) ) ).

% minus_le_iff
tff(fact_7409_minus__le__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),A2) ) ).

% minus_le_iff
tff(fact_7410_minus__le__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),A2) ) ).

% minus_le_iff
tff(fact_7411_le__minus__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(uminus_uminus_real,A2)) ) ).

% le_minus_iff
tff(fact_7412_le__minus__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),uminus1351360451143612070nteger(B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),uminus1351360451143612070nteger(A2)) ) ).

% le_minus_iff
tff(fact_7413_le__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(uminus_uminus_rat,A2)) ) ).

% le_minus_iff
tff(fact_7414_le__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(uminus_uminus_int,B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(uminus_uminus_int,A2)) ) ).

% le_minus_iff
tff(fact_7415_finite__if__eq__beyond__finite,axiom,
    ! [S: set_int,S4: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => finite6197958912794628473et_int(collect_set_int(aa_set5425576183772390127_int_o(aTP_Lamm_eu(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7416_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Code_integer,S4: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aa_set4044230130517860621eger_o(aTP_Lamm_ev(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7417_finite__if__eq__beyond__finite,axiom,
    ! [S: set_complex,S4: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => finite6551019134538273531omplex(collect_set_complex(aa_set8643480048548153067plex_o(aTP_Lamm_ew(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7418_finite__if__eq__beyond__finite,axiom,
    ! [S: set_nat,S4: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => finite1152437895449049373et_nat(collect_set_nat(aa_set7861427066675100071_nat_o(aTP_Lamm_ex(S),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_7419_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(L,plus_p5714425477246183910nteger(U,one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_7420_minus__equation__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = B2 )
    <=> ( aa_real_real(uminus_uminus_real,B2) = A2 ) ) ).

% minus_equation_iff
tff(fact_7421_minus__equation__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = B2 )
    <=> ( aa_int_int(uminus_uminus_int,B2) = A2 ) ) ).

% minus_equation_iff
tff(fact_7422_minus__equation__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A2) = B2 )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B2) = A2 ) ) ).

% minus_equation_iff
tff(fact_7423_minus__equation__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger(A2) = B2 )
    <=> ( uminus1351360451143612070nteger(B2) = A2 ) ) ).

% minus_equation_iff
tff(fact_7424_minus__equation__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A2) = B2 )
    <=> ( aa_rat_rat(uminus_uminus_rat,B2) = A2 ) ) ).

% minus_equation_iff
tff(fact_7425_equation__minus__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = aa_real_real(uminus_uminus_real,B2) )
    <=> ( B2 = aa_real_real(uminus_uminus_real,A2) ) ) ).

% equation_minus_iff
tff(fact_7426_equation__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(uminus_uminus_int,B2) )
    <=> ( B2 = aa_int_int(uminus_uminus_int,A2) ) ) ).

% equation_minus_iff
tff(fact_7427_equation__minus__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 = aa_complex_complex(uminus1482373934393186551omplex,B2) )
    <=> ( B2 = aa_complex_complex(uminus1482373934393186551omplex,A2) ) ) ).

% equation_minus_iff
tff(fact_7428_equation__minus__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 = uminus1351360451143612070nteger(B2) )
    <=> ( B2 = uminus1351360451143612070nteger(A2) ) ) ).

% equation_minus_iff
tff(fact_7429_equation__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(uminus_uminus_rat,B2) )
    <=> ( B2 = aa_rat_rat(uminus_uminus_rat,A2) ) ) ).

% equation_minus_iff
tff(fact_7430_numeral__neq__neg__numeral,axiom,
    ! [M: num,Nb: num] : aa_num_real(numeral_numeral_real,M) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_7431_numeral__neq__neg__numeral,axiom,
    ! [M: num,Nb: num] : aa_num_int(numeral_numeral_int,M) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_7432_numeral__neq__neg__numeral,axiom,
    ! [M: num,Nb: num] : numera6690914467698888265omplex(M) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_7433_numeral__neq__neg__numeral,axiom,
    ! [M: num,Nb: num] : numera6620942414471956472nteger(M) != uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_7434_numeral__neq__neg__numeral,axiom,
    ! [M: num,Nb: num] : aa_num_rat(numeral_numeral_rat,M) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_7435_neg__numeral__neq__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M)) != aa_num_real(numeral_numeral_real,Nb) ).

% neg_numeral_neq_numeral
tff(fact_7436_neg__numeral__neq__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M)) != aa_num_int(numeral_numeral_int,Nb) ).

% neg_numeral_neq_numeral
tff(fact_7437_neg__numeral__neq__numeral,axiom,
    ! [M: num,Nb: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(M)) != numera6690914467698888265omplex(Nb) ).

% neg_numeral_neq_numeral
tff(fact_7438_neg__numeral__neq__numeral,axiom,
    ! [M: num,Nb: num] : uminus1351360451143612070nteger(numera6620942414471956472nteger(M)) != numera6620942414471956472nteger(Nb) ).

% neg_numeral_neq_numeral
tff(fact_7439_neg__numeral__neq__numeral,axiom,
    ! [M: num,Nb: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M)) != aa_num_rat(numeral_numeral_rat,Nb) ).

% neg_numeral_neq_numeral
tff(fact_7440_minus__less__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),A2) ) ).

% minus_less_iff
tff(fact_7441_minus__less__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B2)),A2) ) ).

% minus_less_iff
tff(fact_7442_minus__less__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A2)),B2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B2)),A2) ) ).

% minus_less_iff
tff(fact_7443_minus__less__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),A2) ) ).

% minus_less_iff
tff(fact_7444_less__minus__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(uminus_uminus_real,B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(uminus_uminus_real,A2)) ) ).

% less_minus_iff
tff(fact_7445_less__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(uminus_uminus_int,B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(uminus_uminus_int,A2)) ) ).

% less_minus_iff
tff(fact_7446_less__minus__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),uminus1351360451143612070nteger(B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),uminus1351360451143612070nteger(A2)) ) ).

% less_minus_iff
tff(fact_7447_less__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(uminus_uminus_rat,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(uminus_uminus_rat,A2)) ) ).

% less_minus_iff
tff(fact_7448_verit__negate__coefficient_I2_J,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2)) ) ).

% verit_negate_coefficient(2)
tff(fact_7449_verit__negate__coefficient_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2)) ) ).

% verit_negate_coefficient(2)
tff(fact_7450_verit__negate__coefficient_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(B2)),uminus1351360451143612070nteger(A2)) ) ).

% verit_negate_coefficient(2)
tff(fact_7451_verit__negate__coefficient_I2_J,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A2)) ) ).

% verit_negate_coefficient(2)
tff(fact_7452_is__num__normalize_I8_J,axiom,
    ! [A2: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2)) ).

% is_num_normalize(8)
tff(fact_7453_is__num__normalize_I8_J,axiom,
    ! [A2: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A2),B2)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2)) ).

% is_num_normalize(8)
tff(fact_7454_is__num__normalize_I8_J,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A2),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B2)),aa_complex_complex(uminus1482373934393186551omplex,A2)) ).

% is_num_normalize(8)
tff(fact_7455_is__num__normalize_I8_J,axiom,
    ! [A2: code_integer,B2: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A2,B2)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(B2),uminus1351360451143612070nteger(A2)) ).

% is_num_normalize(8)
tff(fact_7456_is__num__normalize_I8_J,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A2),B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A2)) ).

% is_num_normalize(8)
tff(fact_7457_add_Oinverse__distrib__swap,axiom,
    ! [A2: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A2)) ).

% add.inverse_distrib_swap
tff(fact_7458_add_Oinverse__distrib__swap,axiom,
    ! [A2: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(plus_plus_int(A2),B2)) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A2)) ).

% add.inverse_distrib_swap
tff(fact_7459_add_Oinverse__distrib__swap,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A2),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B2)),aa_complex_complex(uminus1482373934393186551omplex,A2)) ).

% add.inverse_distrib_swap
tff(fact_7460_add_Oinverse__distrib__swap,axiom,
    ! [A2: code_integer,B2: code_integer] : uminus1351360451143612070nteger(plus_p5714425477246183910nteger(A2,B2)) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(B2),uminus1351360451143612070nteger(A2)) ).

% add.inverse_distrib_swap
tff(fact_7461_add_Oinverse__distrib__swap,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(plus_plus_rat(A2),B2)) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A2)) ).

% add.inverse_distrib_swap
tff(fact_7462_group__cancel_Oneg1,axiom,
    ! [A: real,K: real,A2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A2) )
     => ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A2)) ) ) ).

% group_cancel.neg1
tff(fact_7463_group__cancel_Oneg1,axiom,
    ! [A: int,K: int,A2: int] :
      ( ( A = aa_int_int(plus_plus_int(K),A2) )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A2)) ) ) ).

% group_cancel.neg1
tff(fact_7464_group__cancel_Oneg1,axiom,
    ! [A: complex,K: complex,A2: complex] :
      ( ( A = aa_complex_complex(plus_plus_complex(K),A2) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A2)) ) ) ).

% group_cancel.neg1
tff(fact_7465_group__cancel_Oneg1,axiom,
    ! [A: code_integer,K: code_integer,A2: code_integer] :
      ( ( A = plus_p5714425477246183910nteger(K,A2) )
     => ( uminus1351360451143612070nteger(A) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(K),uminus1351360451143612070nteger(A2)) ) ) ).

% group_cancel.neg1
tff(fact_7466_group__cancel_Oneg1,axiom,
    ! [A: rat,K: rat,A2: rat] :
      ( ( A = aa_rat_rat(plus_plus_rat(K),A2) )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A2)) ) ) ).

% group_cancel.neg1
tff(fact_7467_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_7468_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_7469_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_7470_one__neq__neg__one,axiom,
    one_one_Code_integer != uminus1351360451143612070nteger(one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_7471_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_7472_minus__mult__commute,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A2)),B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(uminus_uminus_real,B2)) ).

% minus_mult_commute
tff(fact_7473_minus__mult__commute,axiom,
    ! [A2: int,B2: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(times_times_int(A2),aa_int_int(uminus_uminus_int,B2)) ).

% minus_mult_commute
tff(fact_7474_minus__mult__commute,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),B2) = aa_complex_complex(times_times_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% minus_mult_commute
tff(fact_7475_minus__mult__commute,axiom,
    ! [A2: code_integer,B2: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(A2),B2) = times_3573771949741848930nteger(A2,uminus1351360451143612070nteger(B2)) ).

% minus_mult_commute
tff(fact_7476_minus__mult__commute,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,A2)),B2) = aa_rat_rat(times_times_rat(A2),aa_rat_rat(uminus_uminus_rat,B2)) ).

% minus_mult_commute
tff(fact_7477_square__eq__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),B2) )
    <=> ( ( A2 = B2 )
        | ( A2 = aa_real_real(uminus_uminus_real,B2) ) ) ) ).

% square_eq_iff
tff(fact_7478_square__eq__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(times_times_int(A2),A2) = aa_int_int(times_times_int(B2),B2) )
    <=> ( ( A2 = B2 )
        | ( A2 = aa_int_int(uminus_uminus_int,B2) ) ) ) ).

% square_eq_iff
tff(fact_7479_square__eq__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A2),A2) = aa_complex_complex(times_times_complex(B2),B2) )
    <=> ( ( A2 = B2 )
        | ( A2 = aa_complex_complex(uminus1482373934393186551omplex,B2) ) ) ) ).

% square_eq_iff
tff(fact_7480_square__eq__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger(A2,A2) = times_3573771949741848930nteger(B2,B2) )
    <=> ( ( A2 = B2 )
        | ( A2 = uminus1351360451143612070nteger(B2) ) ) ) ).

% square_eq_iff
tff(fact_7481_square__eq__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(times_times_rat(A2),A2) = aa_rat_rat(times_times_rat(B2),B2) )
    <=> ( ( A2 = B2 )
        | ( A2 = aa_rat_rat(uminus_uminus_rat,B2) ) ) ) ).

% square_eq_iff
tff(fact_7482_minus__diff__minus,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) ).

% minus_diff_minus
tff(fact_7483_minus__diff__minus,axiom,
    ! [A2: int,B2: int] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,A2)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(minus_minus_int(A2),B2)) ).

% minus_diff_minus
tff(fact_7484_minus__diff__minus,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A2),B2)) ).

% minus_diff_minus
tff(fact_7485_minus__diff__minus,axiom,
    ! [A2: code_integer,B2: code_integer] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(A2),uminus1351360451143612070nteger(B2)) = uminus1351360451143612070nteger(minus_8373710615458151222nteger(A2,B2)) ).

% minus_diff_minus
tff(fact_7486_minus__diff__minus,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A2),B2)) ).

% minus_diff_minus
tff(fact_7487_minus__diff__commute,axiom,
    ! [B2: real,A2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B2)),A2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A2)),B2) ).

% minus_diff_commute
tff(fact_7488_minus__diff__commute,axiom,
    ! [B2: int,A2: int] : aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,B2)),A2) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,A2)),B2) ).

% minus_diff_commute
tff(fact_7489_minus__diff__commute,axiom,
    ! [B2: complex,A2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B2)),A2) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),B2) ).

% minus_diff_commute
tff(fact_7490_minus__diff__commute,axiom,
    ! [B2: code_integer,A2: code_integer] : minus_8373710615458151222nteger(uminus1351360451143612070nteger(B2),A2) = minus_8373710615458151222nteger(uminus1351360451143612070nteger(A2),B2) ).

% minus_diff_commute
tff(fact_7491_minus__diff__commute,axiom,
    ! [B2: rat,A2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B2)),A2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A2)),B2) ).

% minus_diff_commute
tff(fact_7492_minus__divide__left,axiom,
    ! [A2: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A2)),B2) ).

% minus_divide_left
tff(fact_7493_minus__divide__left,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A2)),B2) ).

% minus_divide_left
tff(fact_7494_minus__divide__left,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A2),B2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A2)),B2) ).

% minus_divide_left
tff(fact_7495_minus__divide__divide,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) ).

% minus_divide_divide
tff(fact_7496_minus__divide__divide,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(divide1717551699836669952omplex(A2),B2) ).

% minus_divide_divide
tff(fact_7497_minus__divide__divide,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(divide_divide_rat(A2),B2) ).

% minus_divide_divide
tff(fact_7498_minus__divide__right,axiom,
    ! [A2: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(uminus_uminus_real,B2)) ).

% minus_divide_right
tff(fact_7499_minus__divide__right,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) = aa_complex_complex(divide1717551699836669952omplex(A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% minus_divide_right
tff(fact_7500_minus__divide__right,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A2),B2)) = aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(uminus_uminus_rat,B2)) ).

% minus_divide_right
tff(fact_7501_div__minus__right,axiom,
    ! [A2: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2) ).

% div_minus_right
tff(fact_7502_div__minus__right,axiom,
    ! [A2: code_integer,B2: code_integer] : divide6298287555418463151nteger(A2,uminus1351360451143612070nteger(B2)) = divide6298287555418463151nteger(uminus1351360451143612070nteger(A2),B2) ).

% div_minus_right
tff(fact_7503_finite__nat__set__iff__bounded,axiom,
    ! [N5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N5)
    <=> ? [M8: nat] :
        ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),N5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),M8) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_7504_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,Nb: nat] :
      ( ! [X3: nat] :
          ( aa_set_nat_o(member_nat(X3),N5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Nb) )
     => aa_set_nat_o(finite_finite_nat,N5) ) ).

% bounded_nat_set_is_finite
tff(fact_7505_mod__minus__right,axiom,
    ! [A2: int,B2: int] : aa_int_int(modulo_modulo_int(A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A2)),B2)) ).

% mod_minus_right
tff(fact_7506_mod__minus__right,axiom,
    ! [A2: code_integer,B2: code_integer] : modulo364778990260209775nteger(A2,uminus1351360451143612070nteger(B2)) = uminus1351360451143612070nteger(modulo364778990260209775nteger(uminus1351360451143612070nteger(A2),B2)) ).

% mod_minus_right
tff(fact_7507_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A2: int,B2: int,A4: int] :
      ( ( aa_int_int(modulo_modulo_int(A2),B2) = aa_int_int(modulo_modulo_int(A4),B2) )
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A4)),B2) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_7508_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A2: code_integer,B2: code_integer,A4: code_integer] :
      ( ( modulo364778990260209775nteger(A2,B2) = modulo364778990260209775nteger(A4,B2) )
     => ( modulo364778990260209775nteger(uminus1351360451143612070nteger(A2),B2) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A4),B2) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_7509_mod__minus__eq,axiom,
    ! [A2: int,B2: int] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A2),B2))),B2) = aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A2)),B2) ).

% mod_minus_eq
tff(fact_7510_mod__minus__eq,axiom,
    ! [A2: code_integer,B2: code_integer] : modulo364778990260209775nteger(uminus1351360451143612070nteger(modulo364778990260209775nteger(A2,B2)),B2) = modulo364778990260209775nteger(uminus1351360451143612070nteger(A2),B2) ).

% mod_minus_eq
tff(fact_7511_finite__nat__set__iff__bounded__le,axiom,
    ! [N5: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,N5)
    <=> ? [M8: nat] :
        ! [X2: nat] :
          ( aa_set_nat_o(member_nat(X2),N5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),M8) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_7512_finite__list,axiom,
    ! [A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ? [Xs2: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs2) = A ) ).

% finite_list
tff(fact_7513_finite__list,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ? [Xs2: list_real] : set_real2(Xs2) = A ) ).

% finite_list
tff(fact_7514_finite__list,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ? [Xs2: list_nat] : set_nat2(Xs2) = A ) ).

% finite_list
tff(fact_7515_finite__list,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ? [Xs2: list_int] : set_int2(Xs2) = A ) ).

% finite_list
tff(fact_7516_finite__list,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ? [Xs2: list_Code_integer] : set_Code_integer2(Xs2) = A ) ).

% finite_list
tff(fact_7517_finite__list,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ? [Xs2: list_complex] : set_complex2(Xs2) = A ) ).

% finite_list
tff(fact_7518_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K)) ).

% of_int_neg_numeral
tff(fact_7519_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_neg_numeral
tff(fact_7520_of__int__neg__numeral,axiom,
    ! [K: num] : ring_17405671764205052669omplex(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K)) ).

% of_int_neg_numeral
tff(fact_7521_of__int__neg__numeral,axiom,
    ! [K: num] : ring_18347121197199848620nteger(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = uminus1351360451143612070nteger(numera6620942414471956472nteger(K)) ).

% of_int_neg_numeral
tff(fact_7522_of__int__neg__numeral,axiom,
    ! [K: num] : aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K)) ).

% of_int_neg_numeral
tff(fact_7523_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ey(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_7524_finite__less__ub,axiom,
    ! [F2: fun_nat_nat,U: nat] :
      ( ! [N: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),aa_nat_nat(F2,N))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ez(F2),U))) ) ).

% finite_less_ub
tff(fact_7525_finite__lists__length__eq,axiom,
    ! [A: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_fa(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7526_finite__lists__length__eq,axiom,
    ! [A: set_Code_integer,Nb: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_fb(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7527_finite__lists__length__eq,axiom,
    ! [A: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_fc(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7528_finite__lists__length__eq,axiom,
    ! [A: set_VEBT_VEBT,Nb: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_fd(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7529_finite__lists__length__eq,axiom,
    ! [A: set_real,Nb: nat] :
      ( aa_set_real_o(finite_finite_real,A)
     => finite306553202115118035t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_fe(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7530_finite__lists__length__eq,axiom,
    ! [A: set_o,Nb: nat] :
      ( aa_set_o_o(finite_finite_o,A)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_ff(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7531_finite__lists__length__eq,axiom,
    ! [A: set_VEBT_VEBTi,Nb: nat] :
      ( finite6580341952239402572_VEBTi(A)
     => finite5722435864697464412_VEBTi(collec1288327022334394266_VEBTi(aa_nat5816202840298588638EBTi_o(aTP_Lamm_fg(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7532_finite__lists__length__eq,axiom,
    ! [A: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_fh(A),Nb))) ) ).

% finite_lists_length_eq
tff(fact_7533_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% zero_neq_neg_numeral
tff(fact_7534_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% zero_neq_neg_numeral
tff(fact_7535_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_7536_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_z3403309356797280102nteger != uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) ).

% zero_neq_neg_numeral
tff(fact_7537_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% zero_neq_neg_numeral
tff(fact_7538_neg__numeral__le__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_le_numeral
tff(fact_7539_neg__numeral__le__numeral,axiom,
    ! [M: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_le_numeral
tff(fact_7540_neg__numeral__le__numeral,axiom,
    ! [M: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_le_numeral
tff(fact_7541_neg__numeral__le__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_le_numeral
tff(fact_7542_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_7543_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_7544_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_7545_not__numeral__le__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_7546_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_7547_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_7548_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_7549_not__numeral__less__neg__numeral,axiom,
    ! [M: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_7550_neg__numeral__less__numeral,axiom,
    ! [M: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_less_numeral
tff(fact_7551_neg__numeral__less__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_less_numeral
tff(fact_7552_neg__numeral__less__numeral,axiom,
    ! [M: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_less_numeral
tff(fact_7553_neg__numeral__less__numeral,axiom,
    ! [M: num,Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_less_numeral
tff(fact_7554_neg__eq__iff__add__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A2) = B2 )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_7555_neg__eq__iff__add__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(uminus_uminus_int,A2) = B2 )
    <=> ( aa_int_int(plus_plus_int(A2),B2) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_7556_neg__eq__iff__add__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A2) = B2 )
    <=> ( aa_complex_complex(plus_plus_complex(A2),B2) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_7557_neg__eq__iff__add__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger(A2) = B2 )
    <=> ( plus_p5714425477246183910nteger(A2,B2) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_7558_neg__eq__iff__add__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A2) = B2 )
    <=> ( aa_rat_rat(plus_plus_rat(A2),B2) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_7559_eq__neg__iff__add__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = aa_real_real(uminus_uminus_real,B2) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_7560_eq__neg__iff__add__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(uminus_uminus_int,B2) )
    <=> ( aa_int_int(plus_plus_int(A2),B2) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_7561_eq__neg__iff__add__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 = aa_complex_complex(uminus1482373934393186551omplex,B2) )
    <=> ( aa_complex_complex(plus_plus_complex(A2),B2) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_7562_eq__neg__iff__add__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 = uminus1351360451143612070nteger(B2) )
    <=> ( plus_p5714425477246183910nteger(A2,B2) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_7563_eq__neg__iff__add__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(uminus_uminus_rat,B2) )
    <=> ( aa_rat_rat(plus_plus_rat(A2),B2) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_7564_add_Oinverse__unique,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A2) = B2 ) ) ).

% add.inverse_unique
tff(fact_7565_add_Oinverse__unique,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(plus_plus_int(A2),B2) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A2) = B2 ) ) ).

% add.inverse_unique
tff(fact_7566_add_Oinverse__unique,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A2),B2) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A2) = B2 ) ) ).

% add.inverse_unique
tff(fact_7567_add_Oinverse__unique,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger(A2,B2) = zero_z3403309356797280102nteger )
     => ( uminus1351360451143612070nteger(A2) = B2 ) ) ).

% add.inverse_unique
tff(fact_7568_add_Oinverse__unique,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),B2) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A2) = B2 ) ) ).

% add.inverse_unique
tff(fact_7569_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),A2) = zero_zero_real ).

% ab_group_add_class.ab_left_minus
tff(fact_7570_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: int] : aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,A2)),A2) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_7571_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),A2) = zero_zero_complex ).

% ab_group_add_class.ab_left_minus
tff(fact_7572_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: code_integer] : plus_p5714425477246183910nteger(uminus1351360451143612070nteger(A2),A2) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_7573_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: rat] : aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A2)),A2) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_7574_add__eq__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2) = zero_zero_real )
    <=> ( B2 = aa_real_real(uminus_uminus_real,A2) ) ) ).

% add_eq_0_iff
tff(fact_7575_add__eq__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(plus_plus_int(A2),B2) = zero_zero_int )
    <=> ( B2 = aa_int_int(uminus_uminus_int,A2) ) ) ).

% add_eq_0_iff
tff(fact_7576_add__eq__0__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A2),B2) = zero_zero_complex )
    <=> ( B2 = aa_complex_complex(uminus1482373934393186551omplex,A2) ) ) ).

% add_eq_0_iff
tff(fact_7577_add__eq__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger(A2,B2) = zero_z3403309356797280102nteger )
    <=> ( B2 = uminus1351360451143612070nteger(A2) ) ) ).

% add_eq_0_iff
tff(fact_7578_add__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(plus_plus_rat(A2),B2) = zero_zero_rat )
    <=> ( B2 = aa_rat_rat(uminus_uminus_rat,A2) ) ) ).

% add_eq_0_iff
tff(fact_7579_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_7580_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_7581_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_7582_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != uminus1351360451143612070nteger(one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_7583_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_7584_le__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(4)
tff(fact_7585_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_7586_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_7587_le__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(4)
tff(fact_7588_le__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% le_minus_one_simps(2)
tff(fact_7589_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_7590_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_7591_le__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% le_minus_one_simps(2)
tff(fact_7592_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_real(numeral_numeral_real,Nb) != aa_real_real(uminus_uminus_real,one_one_real) ).

% numeral_neq_neg_one
tff(fact_7593_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,Nb) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_7594_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(Nb) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_7595_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6620942414471956472nteger(Nb) != uminus1351360451143612070nteger(one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_7596_numeral__neq__neg__one,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,Nb) != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% numeral_neq_neg_one
tff(fact_7597_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_real != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% one_neq_neg_numeral
tff(fact_7598_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_int != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% one_neq_neg_numeral
tff(fact_7599_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% one_neq_neg_numeral
tff(fact_7600_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_Code_integer != uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb)) ).

% one_neq_neg_numeral
tff(fact_7601_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_rat != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% one_neq_neg_numeral
tff(fact_7602_less__minus__one__simps_I2_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),one_one_real) ).

% less_minus_one_simps(2)
tff(fact_7603_less__minus__one__simps_I2_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) ).

% less_minus_one_simps(2)
tff(fact_7604_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_7605_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_7606_less__minus__one__simps_I4_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(4)
tff(fact_7607_less__minus__one__simps_I4_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(4)
tff(fact_7608_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_7609_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_7610_numeral__times__minus__swap,axiom,
    ! [W: num,Xc: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(uminus_uminus_real,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% numeral_times_minus_swap
tff(fact_7611_numeral__times__minus__swap,axiom,
    ! [W: num,Xc: int] : aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,Xc)) = aa_int_int(times_times_int(Xc),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_7612_numeral__times__minus__swap,axiom,
    ! [W: num,Xc: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,Xc)) = aa_complex_complex(times_times_complex(Xc),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_7613_numeral__times__minus__swap,axiom,
    ! [W: num,Xc: code_integer] : times_3573771949741848930nteger(numera6620942414471956472nteger(W),uminus1351360451143612070nteger(Xc)) = times_3573771949741848930nteger(Xc,uminus1351360451143612070nteger(numera6620942414471956472nteger(W))) ).

% numeral_times_minus_swap
tff(fact_7614_numeral__times__minus__swap,axiom,
    ! [W: num,Xc: rat] : aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,Xc)) = aa_rat_rat(times_times_rat(Xc),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% numeral_times_minus_swap
tff(fact_7615_nonzero__minus__divide__divide,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_7616_nonzero__minus__divide__divide,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(divide1717551699836669952omplex(A2),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_7617_nonzero__minus__divide__divide,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(divide_divide_rat(A2),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_7618_nonzero__minus__divide__right,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_7619_nonzero__minus__divide__right,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) = aa_complex_complex(divide1717551699836669952omplex(A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_7620_nonzero__minus__divide__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A2),B2)) = aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_7621_square__eq__1__iff,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc) = one_one_real )
    <=> ( ( Xc = one_one_real )
        | ( Xc = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_7622_square__eq__1__iff,axiom,
    ! [Xc: int] :
      ( ( aa_int_int(times_times_int(Xc),Xc) = one_one_int )
    <=> ( ( Xc = one_one_int )
        | ( Xc = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_7623_square__eq__1__iff,axiom,
    ! [Xc: complex] :
      ( ( aa_complex_complex(times_times_complex(Xc),Xc) = one_one_complex )
    <=> ( ( Xc = one_one_complex )
        | ( Xc = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_7624_square__eq__1__iff,axiom,
    ! [Xc: code_integer] :
      ( ( times_3573771949741848930nteger(Xc,Xc) = one_one_Code_integer )
    <=> ( ( Xc = one_one_Code_integer )
        | ( Xc = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_7625_square__eq__1__iff,axiom,
    ! [Xc: rat] :
      ( ( aa_rat_rat(times_times_rat(Xc),Xc) = one_one_rat )
    <=> ( ( Xc = one_one_rat )
        | ( Xc = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_7626_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(uminus_uminus_real,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_7627_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: int,B2: int] : aa_int_int(minus_minus_int(A2),B2) = aa_int_int(plus_plus_int(A2),aa_int_int(uminus_uminus_int,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_7628_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(minus_minus_complex(A2),B2) = aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_7629_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A2,B2) = plus_p5714425477246183910nteger(A2,uminus1351360451143612070nteger(B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_7630_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(A2),B2) = aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(uminus_uminus_rat,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_7631_diff__conv__add__uminus,axiom,
    ! [A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),aa_real_real(uminus_uminus_real,B2)) ).

% diff_conv_add_uminus
tff(fact_7632_diff__conv__add__uminus,axiom,
    ! [A2: int,B2: int] : aa_int_int(minus_minus_int(A2),B2) = aa_int_int(plus_plus_int(A2),aa_int_int(uminus_uminus_int,B2)) ).

% diff_conv_add_uminus
tff(fact_7633_diff__conv__add__uminus,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(minus_minus_complex(A2),B2) = aa_complex_complex(plus_plus_complex(A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% diff_conv_add_uminus
tff(fact_7634_diff__conv__add__uminus,axiom,
    ! [A2: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A2,B2) = plus_p5714425477246183910nteger(A2,uminus1351360451143612070nteger(B2)) ).

% diff_conv_add_uminus
tff(fact_7635_diff__conv__add__uminus,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(minus_minus_rat(A2),B2) = aa_rat_rat(plus_plus_rat(A2),aa_rat_rat(uminus_uminus_rat,B2)) ).

% diff_conv_add_uminus
tff(fact_7636_group__cancel_Osub2,axiom,
    ! [B: real,K: real,B2: real,A2: real] :
      ( ( B = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) ) ) ).

% group_cancel.sub2
tff(fact_7637_group__cancel_Osub2,axiom,
    ! [B: int,K: int,B2: int,A2: int] :
      ( ( B = aa_int_int(plus_plus_int(K),B2) )
     => ( aa_int_int(minus_minus_int(A2),B) = aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,K)),aa_int_int(minus_minus_int(A2),B2)) ) ) ).

% group_cancel.sub2
tff(fact_7638_group__cancel_Osub2,axiom,
    ! [B: complex,K: complex,B2: complex,A2: complex] :
      ( ( B = aa_complex_complex(plus_plus_complex(K),B2) )
     => ( aa_complex_complex(minus_minus_complex(A2),B) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A2),B2)) ) ) ).

% group_cancel.sub2
tff(fact_7639_group__cancel_Osub2,axiom,
    ! [B: code_integer,K: code_integer,B2: code_integer,A2: code_integer] :
      ( ( B = plus_p5714425477246183910nteger(K,B2) )
     => ( minus_8373710615458151222nteger(A2,B) = plus_p5714425477246183910nteger(uminus1351360451143612070nteger(K),minus_8373710615458151222nteger(A2,B2)) ) ) ).

% group_cancel.sub2
tff(fact_7640_group__cancel_Osub2,axiom,
    ! [B: rat,K: rat,B2: rat,A2: rat] :
      ( ( B = aa_rat_rat(plus_plus_rat(K),B2) )
     => ( aa_rat_rat(minus_minus_rat(A2),B) = aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(A2),B2)) ) ) ).

% group_cancel.sub2
tff(fact_7641_dvd__neg__div,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A2)),B2) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) ) ) ).

% dvd_neg_div
tff(fact_7642_dvd__neg__div,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% dvd_neg_div
tff(fact_7643_dvd__neg__div,axiom,
    ! [B2: complex,A2: complex] :
      ( aa_complex_o(dvd_dvd_complex(B2),A2)
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A2)),B2) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) ) ) ).

% dvd_neg_div
tff(fact_7644_dvd__neg__div,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
     => ( divide6298287555418463151nteger(uminus1351360451143612070nteger(A2),B2) = uminus1351360451143612070nteger(divide6298287555418463151nteger(A2,B2)) ) ) ).

% dvd_neg_div
tff(fact_7645_dvd__neg__div,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A2)),B2) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A2),B2)) ) ) ).

% dvd_neg_div
tff(fact_7646_dvd__div__neg,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) ) ) ).

% dvd_div_neg
tff(fact_7647_dvd__div__neg,axiom,
    ! [B2: int,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% dvd_div_neg
tff(fact_7648_dvd__div__neg,axiom,
    ! [B2: complex,A2: complex] :
      ( aa_complex_o(dvd_dvd_complex(B2),A2)
     => ( aa_complex_complex(divide1717551699836669952omplex(A2),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) ) ) ).

% dvd_div_neg
tff(fact_7649_dvd__div__neg,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A2)
     => ( divide6298287555418463151nteger(A2,uminus1351360451143612070nteger(B2)) = uminus1351360451143612070nteger(divide6298287555418463151nteger(A2,B2)) ) ) ).

% dvd_div_neg
tff(fact_7650_dvd__div__neg,axiom,
    ! [B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A2)
     => ( aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A2),B2)) ) ) ).

% dvd_div_neg
tff(fact_7651_infinite__Icc,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ~ aa_set_rat_o(finite_finite_rat,set_or633870826150836451st_rat(A2,B2)) ) ).

% infinite_Icc
tff(fact_7652_infinite__Icc,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ~ aa_set_real_o(finite_finite_real,set_or1222579329274155063t_real(A2,B2)) ) ).

% infinite_Icc
tff(fact_7653_infinite__Ico,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ~ aa_set_real_o(finite_finite_real,set_or66887138388493659n_real(A2,B2)) ) ).

% infinite_Ico
tff(fact_7654_infinite__Ico,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
     => ~ aa_set_rat_o(finite_finite_rat,set_or4029947393144176647an_rat(A2,B2)) ) ).

% infinite_Ico
tff(fact_7655_finite__lists__length__le,axiom,
    ! [A: set_int,Nb: nat] :
      ( aa_set_int_o(finite_finite_int,A)
     => finite3922522038869484883st_int(collect_list_int(aa_nat3638697132715757351_int_o(aTP_Lamm_fi(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_7656_finite__lists__length__le,axiom,
    ! [A: set_Code_integer,Nb: nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => finite1283093830868386564nteger(collec3483841146883906114nteger(aa_nat2697030861539126eger_o(aTP_Lamm_fj(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_7657_finite__lists__length__le,axiom,
    ! [A: set_complex,Nb: nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => finite8712137658972009173omplex(collect_list_complex(aa_nat5940423528545770789plex_o(aTP_Lamm_fk(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_7658_finite__lists__length__le,axiom,
    ! [A: set_VEBT_VEBT,Nb: nat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => finite3004134309566078307T_VEBT(collec5608196760682091941T_VEBT(aa_nat1274389614848952367VEBT_o(aTP_Lamm_fl(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_7659_finite__lists__length__le,axiom,
    ! [A: set_real,Nb: nat] :
      ( aa_set_real_o(finite_finite_real,A)
     => finite306553202115118035t_real(collect_list_real(aa_nat3629845878562903719real_o(aTP_Lamm_fm(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_7660_finite__lists__length__le,axiom,
    ! [A: set_o,Nb: nat] :
      ( aa_set_o_o(finite_finite_o,A)
     => finite_finite_list_o(collect_list_o(aa_nat_fun_list_o_o(aTP_Lamm_fn(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_7661_finite__lists__length__le,axiom,
    ! [A: set_VEBT_VEBTi,Nb: nat] :
      ( finite6580341952239402572_VEBTi(A)
     => finite5722435864697464412_VEBTi(collec1288327022334394266_VEBTi(aa_nat5816202840298588638EBTi_o(aTP_Lamm_fo(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_7662_finite__lists__length__le,axiom,
    ! [A: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => finite8100373058378681591st_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_fp(A),Nb))) ) ).

% finite_lists_length_le
tff(fact_7663_real__minus__mult__self__le,axiom,
    ! [U: real,Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc)) ).

% real_minus_mult_self_le
tff(fact_7664_minus__real__def,axiom,
    ! [Xc: real,Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Ya) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(uminus_uminus_real,Ya)) ).

% minus_real_def
tff(fact_7665_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7666_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7667_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7668_of__int__of__nat,axiom,
    ! [K: int] :
      ring_17405671764205052669omplex(K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7669_of__int__of__nat,axiom,
    ! [K: int] :
      ring_18347121197199848620nteger(K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),uminus1351360451143612070nteger(aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_Code_integer(semiri4939895301339042750nteger,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7670_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_7671_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) ).

% not_zero_le_neg_numeral
tff(fact_7672_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_7673_not__zero__le__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_le_neg_numeral
tff(fact_7674_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_le_zero
tff(fact_7675_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_le_zero
tff(fact_7676_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_le_zero
tff(fact_7677_neg__numeral__le__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_le_zero
tff(fact_7678_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_7679_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_7680_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) ).

% not_zero_less_neg_numeral
tff(fact_7681_not__zero__less__neg__numeral,axiom,
    ! [Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_zero_less_neg_numeral
tff(fact_7682_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))),zero_zero_real) ).

% neg_numeral_less_zero
tff(fact_7683_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),zero_zero_int) ).

% neg_numeral_less_zero
tff(fact_7684_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))),zero_z3403309356797280102nteger) ).

% neg_numeral_less_zero
tff(fact_7685_neg__numeral__less__zero,axiom,
    ! [Nb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))),zero_zero_rat) ).

% neg_numeral_less_zero
tff(fact_7686_le__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% le_minus_one_simps(3)
tff(fact_7687_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_7688_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_7689_le__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% le_minus_one_simps(3)
tff(fact_7690_le__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% le_minus_one_simps(1)
tff(fact_7691_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_7692_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_7693_le__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% le_minus_one_simps(1)
tff(fact_7694_less__minus__one__simps_I1_J,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),zero_zero_real) ).

% less_minus_one_simps(1)
tff(fact_7695_less__minus__one__simps_I1_J,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),zero_zero_int) ).

% less_minus_one_simps(1)
tff(fact_7696_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_7697_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_7698_less__minus__one__simps_I3_J,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,one_one_real)) ).

% less_minus_one_simps(3)
tff(fact_7699_less__minus__one__simps_I3_J,axiom,
    ~ aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% less_minus_one_simps(3)
tff(fact_7700_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_7701_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_7702_neg__numeral__le__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real) ).

% neg_numeral_le_one
tff(fact_7703_neg__numeral__le__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_7704_neg__numeral__le__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_7705_neg__numeral__le__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_le_one
tff(fact_7706_neg__one__le__numeral,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M)) ).

% neg_one_le_numeral
tff(fact_7707_neg__one__le__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),numera6620942414471956472nteger(M)) ).

% neg_one_le_numeral
tff(fact_7708_neg__one__le__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M)) ).

% neg_one_le_numeral
tff(fact_7709_neg__one__le__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_le_numeral
tff(fact_7710_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_7711_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_7712_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_7713_neg__numeral__le__neg__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_7714_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_7715_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_7716_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_7717_not__numeral__le__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_7718_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))) ).

% not_one_le_neg_numeral
tff(fact_7719_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),uminus1351360451143612070nteger(numera6620942414471956472nteger(M))) ).

% not_one_le_neg_numeral
tff(fact_7720_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))) ).

% not_one_le_neg_numeral
tff(fact_7721_not__one__le__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_le_neg_numeral
tff(fact_7722_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_7723_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_7724_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),uminus1351360451143612070nteger(numera6620942414471956472nteger(M))) ).

% not_neg_one_less_neg_numeral
tff(fact_7725_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))) ).

% not_neg_one_less_neg_numeral
tff(fact_7726_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))) ).

% not_one_less_neg_numeral
tff(fact_7727_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))) ).

% not_one_less_neg_numeral
tff(fact_7728_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),uminus1351360451143612070nteger(numera6620942414471956472nteger(M))) ).

% not_one_less_neg_numeral
tff(fact_7729_not__one__less__neg__numeral,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))) ).

% not_one_less_neg_numeral
tff(fact_7730_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,M)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_7731_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_7732_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(M)),uminus1351360451143612070nteger(one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_7733_not__numeral__less__neg__one,axiom,
    ! [M: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,M)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_7734_neg__one__less__numeral,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,M)) ).

% neg_one_less_numeral
tff(fact_7735_neg__one__less__numeral,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,M)) ).

% neg_one_less_numeral
tff(fact_7736_neg__one__less__numeral,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(one_one_Code_integer)),numera6620942414471956472nteger(M)) ).

% neg_one_less_numeral
tff(fact_7737_neg__one__less__numeral,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_num_rat(numeral_numeral_rat,M)) ).

% neg_one_less_numeral
tff(fact_7738_neg__numeral__less__one,axiom,
    ! [M: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,M))),one_one_real) ).

% neg_numeral_less_one
tff(fact_7739_neg__numeral__less__one,axiom,
    ! [M: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),one_one_int) ).

% neg_numeral_less_one
tff(fact_7740_neg__numeral__less__one,axiom,
    ! [M: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(numera6620942414471956472nteger(M))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_7741_neg__numeral__less__one,axiom,
    ! [M: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,M))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_7742_uminus__numeral__One,axiom,
    aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% uminus_numeral_One
tff(fact_7743_uminus__numeral__One,axiom,
    aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% uminus_numeral_One
tff(fact_7744_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_7745_uminus__numeral__One,axiom,
    uminus1351360451143612070nteger(numera6620942414471956472nteger(one)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_7746_uminus__numeral__One,axiom,
    aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% uminus_numeral_One
tff(fact_7747_mult__1s__ring__1_I1_J,axiom,
    ! [B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))),B2) = aa_real_real(uminus_uminus_real,B2) ).

% mult_1s_ring_1(1)
tff(fact_7748_mult__1s__ring__1_I1_J,axiom,
    ! [B2: int] : aa_int_int(times_times_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B2) = aa_int_int(uminus_uminus_int,B2) ).

% mult_1s_ring_1(1)
tff(fact_7749_mult__1s__ring__1_I1_J,axiom,
    ! [B2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B2) = aa_complex_complex(uminus1482373934393186551omplex,B2) ).

% mult_1s_ring_1(1)
tff(fact_7750_mult__1s__ring__1_I1_J,axiom,
    ! [B2: code_integer] : times_3573771949741848930nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(one)),B2) = uminus1351360451143612070nteger(B2) ).

% mult_1s_ring_1(1)
tff(fact_7751_mult__1s__ring__1_I1_J,axiom,
    ! [B2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B2) = aa_rat_rat(uminus_uminus_rat,B2) ).

% mult_1s_ring_1(1)
tff(fact_7752_mult__1s__ring__1_I2_J,axiom,
    ! [B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))) = aa_real_real(uminus_uminus_real,B2) ).

% mult_1s_ring_1(2)
tff(fact_7753_mult__1s__ring__1_I2_J,axiom,
    ! [B2: int] : aa_int_int(times_times_int(B2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B2) ).

% mult_1s_ring_1(2)
tff(fact_7754_mult__1s__ring__1_I2_J,axiom,
    ! [B2: complex] : aa_complex_complex(times_times_complex(B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B2) ).

% mult_1s_ring_1(2)
tff(fact_7755_mult__1s__ring__1_I2_J,axiom,
    ! [B2: code_integer] : times_3573771949741848930nteger(B2,uminus1351360451143612070nteger(numera6620942414471956472nteger(one))) = uminus1351360451143612070nteger(B2) ).

% mult_1s_ring_1(2)
tff(fact_7756_mult__1s__ring__1_I2_J,axiom,
    ! [B2: rat] : aa_rat_rat(times_times_rat(B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B2) ).

% mult_1s_ring_1(2)
tff(fact_7757_divide__eq__minus__1__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B2 != zero_zero_real )
        & ( A2 = aa_real_real(uminus_uminus_real,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_7758_divide__eq__minus__1__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A2),B2) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B2 != zero_zero_complex )
        & ( A2 = aa_complex_complex(uminus1482373934393186551omplex,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_7759_divide__eq__minus__1__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A2),B2) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B2 != zero_zero_rat )
        & ( A2 = aa_rat_rat(uminus_uminus_rat,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_7760_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( C2 = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) = aa_real_real(uminus_uminus_real,A2) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_7761_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: complex,C2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( C2 = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) )
      <=> ( aa_complex_complex(times_times_complex(C2),B2) = aa_complex_complex(uminus1482373934393186551omplex,A2) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_7762_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( C2 = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A2),B2)) )
      <=> ( aa_rat_rat(times_times_rat(C2),B2) = aa_rat_rat(uminus_uminus_rat,A2) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_7763_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) = C2 )
      <=> ( aa_real_real(uminus_uminus_real,A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_7764_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: complex,A2: complex,C2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) = C2 )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A2) = aa_complex_complex(times_times_complex(C2),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_7765_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A2),B2)) = C2 )
      <=> ( aa_rat_rat(uminus_uminus_rat,A2) = aa_rat_rat(times_times_rat(C2),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_7766_minus__divide__eq__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) = A2 )
    <=> $ite(C2 != zero_zero_real,aa_real_real(uminus_uminus_real,B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2),A2 = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_7767_minus__divide__eq__eq,axiom,
    ! [B2: complex,C2: complex,A2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B2),C2)) = A2 )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B2) = aa_complex_complex(times_times_complex(A2),C2),A2 = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_7768_minus__divide__eq__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2)) = A2 )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B2) = aa_rat_rat(times_times_rat(A2),C2),A2 = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_7769_eq__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( A2 = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)) )
    <=> $ite(C2 != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2) = aa_real_real(uminus_uminus_real,B2),A2 = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_7770_eq__minus__divide__eq,axiom,
    ! [A2: complex,B2: complex,C2: complex] :
      ( ( A2 = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B2),C2)) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(A2),C2) = aa_complex_complex(uminus1482373934393186551omplex,B2),A2 = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_7771_eq__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2 = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2)) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(times_times_rat(A2),C2) = aa_rat_rat(uminus_uminus_rat,B2),A2 = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_7772_power__minus,axiom,
    ! [A2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ).

% power_minus
tff(fact_7773_power__minus,axiom,
    ! [A2: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),Nb) = aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(A2),Nb)) ).

% power_minus
tff(fact_7774_power__minus,axiom,
    ! [A2: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(power_power_complex(A2),Nb)) ).

% power_minus
tff(fact_7775_power__minus,axiom,
    ! [A2: code_integer,Nb: nat] : power_8256067586552552935nteger(uminus1351360451143612070nteger(A2),Nb) = times_3573771949741848930nteger(power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),Nb),power_8256067586552552935nteger(A2,Nb)) ).

% power_minus
tff(fact_7776_power__minus,axiom,
    ! [A2: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A2)),Nb) = aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(A2),Nb)) ).

% power_minus
tff(fact_7777_power__minus__Bit0,axiom,
    ! [Xc: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_7778_power__minus__Bit0,axiom,
    ! [Xc: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_7779_power__minus__Bit0,axiom,
    ! [Xc: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(power_power_complex(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_7780_power__minus__Bit0,axiom,
    ! [Xc: code_integer,K: num] : power_8256067586552552935nteger(uminus1351360451143612070nteger(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_7781_power__minus__Bit0,axiom,
    ! [Xc: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_7782_power__minus__Bit1,axiom,
    ! [Xc: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_7783_power__minus__Bit1,axiom,
    ! [Xc: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_7784_power__minus__Bit1,axiom,
    ! [Xc: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_7785_power__minus__Bit1,axiom,
    ! [Xc: code_integer,K: num] : power_8256067586552552935nteger(uminus1351360451143612070nteger(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = uminus1351360451143612070nteger(power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_7786_power__minus__Bit1,axiom,
    ! [Xc: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_7787_norm__uminus__minus,axiom,
    ! [Xc: real,Ya: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xc)),Ya)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)) ).

% norm_uminus_minus
tff(fact_7788_norm__uminus__minus,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xc)),Ya)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xc),Ya)) ).

% norm_uminus_minus
tff(fact_7789_real__add__less__0__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(uminus_uminus_real,Xc)) ) ).

% real_add_less_0_iff
tff(fact_7790_real__0__less__add__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xc)),Ya) ) ).

% real_0_less_add_iff
tff(fact_7791_real__add__le__0__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(uminus_uminus_real,Xc)) ) ).

% real_add_le_0_iff
tff(fact_7792_real__0__le__add__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xc)),Ya) ) ).

% real_0_le_add_iff
tff(fact_7793_tanh__real__gt__neg1,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,Xc)) ).

% tanh_real_gt_neg1
tff(fact_7794_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ed,M))) ) ).

% finite_divisors_nat
tff(fact_7795_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N5) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_7796_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
     => aa_set_nat_o(finite_finite_nat,N5) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_7797_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) )
    <=> $ite(C2 != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C2) = B2,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_7798_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: complex,C2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(B2),C2) )
    <=> $ite(C2 != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C2) = B2,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_7799_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(B2),C2) )
    <=> $ite(C2 != zero_zero_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C2) = B2,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_7800_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(C2 != zero_zero_real,B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% divide_eq_eq_numeral(2)
tff(fact_7801_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: complex,C2: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C2) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> $ite(C2 != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(2)
tff(fact_7802_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B2),C2) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(C2 != zero_zero_rat,B2 = aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(2)
tff(fact_7803_pos__minus__divide__less__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_7804_pos__minus__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2))),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(times_times_rat(A2),C2)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_7805_pos__less__minus__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_7806_pos__less__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_7807_neg__minus__divide__less__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_7808_neg__minus__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2))),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_7809_neg__less__minus__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_7810_neg__less__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(times_times_rat(A2),C2)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_7811_minus__divide__less__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))),A2)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(uminus_uminus_real,B2)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)) ) ) ).

% minus_divide_less_eq
tff(fact_7812_minus__divide__less__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2))),A2)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(times_times_rat(A2),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)) ) ) ).

% minus_divide_less_eq
tff(fact_7813_less__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(uminus_uminus_real,B2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)) ) ) ).

% less_minus_divide_eq
tff(fact_7814_less__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(uminus_uminus_rat,B2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(times_times_rat(A2),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_7815_minus__divide__add__eq__iff,axiom,
    ! [Z: real,Xc: real,Ya: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Z))),Ya) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_7816_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,Xc: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xc),Z))),Ya) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xc)),aa_complex_complex(times_times_complex(Ya),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_7817_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xc),Z))),Ya) = aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,Xc)),aa_rat_rat(times_times_rat(Ya),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_7818_add__divide__eq__if__simps_I3_J,axiom,
    ! [A2: real,Z: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),Z))),B2) = $ite(Z = zero_zero_real,B2,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_7819_add__divide__eq__if__simps_I3_J,axiom,
    ! [A2: complex,Z: complex,B2: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),Z))),B2) = $ite(Z = zero_zero_complex,B2,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(times_times_complex(B2),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_7820_add__divide__eq__if__simps_I3_J,axiom,
    ! [A2: rat,Z: rat,B2: rat] :
      aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A2),Z))),B2) = $ite(Z = zero_zero_rat,B2,aa_rat_rat(divide_divide_rat(aa_rat_rat(plus_plus_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(times_times_rat(B2),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_7821_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,Xc: real,Ya: real] :
      ( ( Z != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Z))),Ya) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_7822_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,Xc: complex,Ya: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xc),Z))),Ya) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xc)),aa_complex_complex(times_times_complex(Ya),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_7823_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,Xc: rat,Ya: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xc),Z))),Ya) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Xc)),aa_rat_rat(times_times_rat(Ya),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_7824_add__divide__eq__if__simps_I5_J,axiom,
    ! [A2: real,Z: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),Z)),B2) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_7825_add__divide__eq__if__simps_I5_J,axiom,
    ! [A2: complex,Z: complex,B2: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A2),Z)),B2) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B2),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A2),aa_complex_complex(times_times_complex(B2),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_7826_add__divide__eq__if__simps_I5_J,axiom,
    ! [A2: rat,Z: rat,B2: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A2),Z)),B2) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B2),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A2),aa_rat_rat(times_times_rat(B2),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_7827_add__divide__eq__if__simps_I6_J,axiom,
    ! [A2: real,Z: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),Z))),B2) = $ite(Z = zero_zero_real,aa_real_real(uminus_uminus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_7828_add__divide__eq__if__simps_I6_J,axiom,
    ! [A2: complex,Z: complex,B2: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A2),Z))),B2) = $ite(Z = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B2),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),aa_complex_complex(times_times_complex(B2),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_7829_add__divide__eq__if__simps_I6_J,axiom,
    ! [A2: rat,Z: rat,B2: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A2),Z))),B2) = $ite(Z = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B2),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(times_times_rat(B2),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_7830_power2__eq__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xc = Ya )
        | ( Xc = aa_real_real(uminus_uminus_real,Ya) ) ) ) ).

% power2_eq_iff
tff(fact_7831_power2__eq__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( ( aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xc = Ya )
        | ( Xc = aa_int_int(uminus_uminus_int,Ya) ) ) ) ).

% power2_eq_iff
tff(fact_7832_power2__eq__iff,axiom,
    ! [Xc: complex,Ya: complex] :
      ( ( aa_nat_complex(power_power_complex(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xc = Ya )
        | ( Xc = aa_complex_complex(uminus1482373934393186551omplex,Ya) ) ) ) ).

% power2_eq_iff
tff(fact_7833_power2__eq__iff,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( ( power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xc = Ya )
        | ( Xc = uminus1351360451143612070nteger(Ya) ) ) ) ).

% power2_eq_iff
tff(fact_7834_power2__eq__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ( aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xc = Ya )
        | ( Xc = aa_rat_rat(uminus_uminus_rat,Ya) ) ) ) ).

% power2_eq_iff
tff(fact_7835_even__minus,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(uminus_uminus_int,A2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2) ) ).

% even_minus
tff(fact_7836_even__minus,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),uminus1351360451143612070nteger(A2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A2) ) ).

% even_minus
tff(fact_7837_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_real_o(finite_finite_real,collect_real(aa_nat_fun_real_o(aTP_Lamm_fq,Nb))) ) ).

% finite_roots_unity
tff(fact_7838_finite__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_fr(Nb))) ) ).

% finite_roots_unity
tff(fact_7839_pos__minus__divide__le__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_7840_pos__minus__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2))),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(times_times_rat(A2),C2)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_7841_pos__le__minus__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_7842_pos__le__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_7843_neg__minus__divide__le__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))),A2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_7844_neg__minus__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2))),A2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_7845_neg__le__minus__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_7846_neg__le__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(times_times_rat(A2),C2)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_7847_minus__divide__le__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))),A2)
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(uminus_uminus_real,B2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)) ) ) ).

% minus_divide_le_eq
tff(fact_7848_minus__divide__le__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2))),A2)
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(times_times_rat(A2),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)) ) ) ).

% minus_divide_le_eq
tff(fact_7849_le__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(uminus_uminus_real,B2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)) ) ) ).

% le_minus_divide_eq
tff(fact_7850_le__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C2)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(A2),C2)),aa_rat_rat(uminus_uminus_rat,B2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(times_times_rat(A2),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_7851_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C2)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C2)),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_7852_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B2),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C2)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_7853_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C2)),B2),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_7854_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),C2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C2)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_7855_power2__eq__1__iff,axiom,
    ! [A2: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( ( A2 = one_one_real )
        | ( A2 = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% power2_eq_1_iff
tff(fact_7856_power2__eq__1__iff,axiom,
    ! [A2: int] :
      ( ( aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( ( A2 = one_one_int )
        | ( A2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_7857_power2__eq__1__iff,axiom,
    ! [A2: complex] :
      ( ( aa_nat_complex(power_power_complex(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
    <=> ( ( A2 = one_one_complex )
        | ( A2 = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% power2_eq_1_iff
tff(fact_7858_power2__eq__1__iff,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( ( A2 = one_one_Code_integer )
        | ( A2 = uminus1351360451143612070nteger(one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_7859_power2__eq__1__iff,axiom,
    ! [A2: rat] :
      ( ( aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( ( A2 = one_one_rat )
        | ( A2 = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_7860_uminus__power__if,axiom,
    ! [A2: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb),aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb))) ).

% uminus_power_if
tff(fact_7861_uminus__power__if,axiom,
    ! [A2: int,Nb: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_int(power_power_int(A2),Nb),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(A2),Nb))) ).

% uminus_power_if
tff(fact_7862_uminus__power__if,axiom,
    ! [A2: complex,Nb: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_complex(power_power_complex(A2),Nb),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(A2),Nb))) ).

% uminus_power_if
tff(fact_7863_uminus__power__if,axiom,
    ! [A2: code_integer,Nb: nat] :
      power_8256067586552552935nteger(uminus1351360451143612070nteger(A2),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),power_8256067586552552935nteger(A2,Nb),uminus1351360451143612070nteger(power_8256067586552552935nteger(A2,Nb))) ).

% uminus_power_if
tff(fact_7864_uminus__power__if,axiom,
    ! [A2: rat,Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A2)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),aa_nat_rat(power_power_rat(A2),Nb),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(A2),Nb))) ).

% uminus_power_if
tff(fact_7865_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7866_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7867_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7868_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7869_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_7870_realpow__square__minus__le,axiom,
    ! [U: real,Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_7871_ln__add__one__self__le__self2,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc))),Xc) ) ).

% ln_add_one_self_le_self2
tff(fact_7872_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C2)),B2),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),zero_zero_real)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_7873_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),aa_rat_rat(divide_divide_rat(B2),C2))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C2)),B2),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_7874_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C2)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))
    <=> $ite(
          aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2),
          aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C2)),
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,C2),zero_zero_real),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))),C2)),B2),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_7875_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),C2)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C2),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C2)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C2),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(times_times_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C2)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_7876_square__le__1,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_7877_square__le__1,axiom,
    ! [Xc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(one_one_Code_integer)),Xc)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xc),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_7878_square__le__1,axiom,
    ! [Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xc)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_7879_square__le__1,axiom,
    ! [Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_7880_minus__power__mult__self,axiom,
    ! [A2: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_7881_minus__power__mult__self,axiom,
    ! [A2: int,Nb: nat] : aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),Nb)) = aa_nat_int(power_power_int(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_7882_minus__power__mult__self,axiom,
    ! [A2: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,A2)),Nb)) = aa_nat_complex(power_power_complex(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_7883_minus__power__mult__self,axiom,
    ! [A2: code_integer,Nb: nat] : times_3573771949741848930nteger(power_8256067586552552935nteger(uminus1351360451143612070nteger(A2),Nb),power_8256067586552552935nteger(uminus1351360451143612070nteger(A2),Nb)) = power_8256067586552552935nteger(A2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_7884_minus__power__mult__self,axiom,
    ! [A2: rat,Nb: nat] : aa_rat_rat(times_times_rat(aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A2)),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A2)),Nb)) = aa_nat_rat(power_power_rat(A2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_7885_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% minus_one_power_iff
tff(fact_7886_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_7887_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) ).

% minus_one_power_iff
tff(fact_7888_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_Code_integer,uminus1351360451143612070nteger(one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_7889_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% minus_one_power_iff
tff(fact_7890_Bernoulli__inequality,axiom,
    ! [Xc: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_7891_ln__one__minus__pos__upper__bound,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xc))),aa_real_real(uminus_uminus_real,Xc)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_7892_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% power_minus1_odd
tff(fact_7893_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_7894_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power_minus1_odd
tff(fact_7895_power__minus1__odd,axiom,
    ! [Nb: nat] : power_8256067586552552935nteger(uminus1351360451143612070nteger(one_one_Code_integer),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% power_minus1_odd
tff(fact_7896_power__minus1__odd,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% power_minus1_odd
tff(fact_7897_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(
        K = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: real,
            l:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_7898_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(
        K = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: int,
            l:= aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_int_int(plus_plus_int(l),one_one_int)) ) ) ) ).

% of_int_code_if
tff(fact_7899_of__int__code__if,axiom,
    ! [K: int] :
      ring_17405671764205052669omplex(K) = $ite(
        K = zero_zero_int,
        zero_zero_complex,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_complex_complex(uminus1482373934393186551omplex,ring_17405671764205052669omplex(aa_int_int(uminus_uminus_int,K))),
          $let(
            l: complex,
            l:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),ring_17405671764205052669omplex(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_complex_complex(plus_plus_complex(l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_7900_of__int__code__if,axiom,
    ! [K: int] :
      ring_18347121197199848620nteger(K) = $ite(
        K = zero_zero_int,
        zero_z3403309356797280102nteger,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          uminus1351360451143612070nteger(ring_18347121197199848620nteger(aa_int_int(uminus_uminus_int,K))),
          $let(
            l: code_integer,
            l:= times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),ring_18347121197199848620nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_7901_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(
        K = zero_zero_int,
        zero_zero_rat,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_rat_rat(uminus_uminus_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: rat,
            l:= aa_rat_rat(times_times_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_rat_rat(plus_plus_rat(l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_7902_finite__Diff__insert,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A,insert_VEBT_VEBT(A2,B)))
    <=> aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A,B)) ) ).

% finite_Diff_insert
tff(fact_7903_finite__Diff__insert,axiom,
    ! [A: set_real,A2: real,B: set_real] :
      ( aa_set_real_o(finite_finite_real,minus_minus_set_real(A,insert_real(A2,B)))
    <=> aa_set_real_o(finite_finite_real,minus_minus_set_real(A,B)) ) ).

% finite_Diff_insert
tff(fact_7904_finite__Diff__insert,axiom,
    ! [A: set_o,A2: $o,B: set_o] :
      ( aa_set_o_o(finite_finite_o,minus_minus_set_o(A,insert_o((A2),B)))
    <=> aa_set_o_o(finite_finite_o,minus_minus_set_o(A,B)) ) ).

% finite_Diff_insert
tff(fact_7905_finite__Diff__insert,axiom,
    ! [A: set_int,A2: int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A,insert_int(A2,B)))
    <=> aa_set_int_o(finite_finite_int,minus_minus_set_int(A,B)) ) ).

% finite_Diff_insert
tff(fact_7906_finite__Diff__insert,axiom,
    ! [A: set_Code_integer,A2: code_integer,B: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A,insert_Code_integer(A2,B)))
    <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A,B)) ) ).

% finite_Diff_insert
tff(fact_7907_finite__Diff__insert,axiom,
    ! [A: set_complex,A2: complex,B: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A,insert_complex(A2,B)))
    <=> aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A,B)) ) ).

% finite_Diff_insert
tff(fact_7908_finite__Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A,insert_nat(A2,B)))
    <=> aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A,B)) ) ).

% finite_Diff_insert
tff(fact_7909_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fs,K))) ).

% finite_Collect_le_nat
tff(fact_7910_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ft,K))) ).

% finite_Collect_less_nat
tff(fact_7911_finite__Collect__subsets,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => finite6197958912794628473et_int(collect_set_int(aTP_Lamm_fu(A))) ) ).

% finite_Collect_subsets
tff(fact_7912_finite__Collect__subsets,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => finite6931041176100689706nteger(collec574505750873337192nteger(aTP_Lamm_fv(A))) ) ).

% finite_Collect_subsets
tff(fact_7913_finite__Collect__subsets,axiom,
    ! [A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => finite6551019134538273531omplex(collect_set_complex(aTP_Lamm_fw(A))) ) ).

% finite_Collect_subsets
tff(fact_7914_finite__Collect__subsets,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => finite1152437895449049373et_nat(collect_set_nat(aTP_Lamm_fx(A))) ) ).

% finite_Collect_subsets
tff(fact_7915_Compl__subset__Compl__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(A)),uminus5710092332889474511et_nat(B))
    <=> aa_set_nat_o(ord_less_eq_set_nat(B),A) ) ).

% Compl_subset_Compl_iff
tff(fact_7916_Compl__anti__mono,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(B)),uminus5710092332889474511et_nat(A)) ) ).

% Compl_anti_mono
tff(fact_7917_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,set_or1266510415728281911st_int(L,U)) ).

% finite_atLeastAtMost_int
tff(fact_7918_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,set_or4662586982721622107an_int(L,U)) ).

% finite_atLeastLessThan_int
tff(fact_7919_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_fy(A2),B2))) ).

% finite_interval_int4
tff(fact_7920_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_7921_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or189985376899183464nteger(L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_7922_finite__insert,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,insert_VEBT_VEBT(A2,A))
    <=> aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A) ) ).

% finite_insert
tff(fact_7923_finite__insert,axiom,
    ! [A2: real,A: set_real] :
      ( aa_set_real_o(finite_finite_real,insert_real(A2,A))
    <=> aa_set_real_o(finite_finite_real,A) ) ).

% finite_insert
tff(fact_7924_finite__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( aa_set_o_o(finite_finite_o,insert_o((A2),A))
    <=> aa_set_o_o(finite_finite_o,A) ) ).

% finite_insert
tff(fact_7925_finite__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,insert_nat(A2,A))
    <=> aa_set_nat_o(finite_finite_nat,A) ) ).

% finite_insert
tff(fact_7926_finite__insert,axiom,
    ! [A2: int,A: set_int] :
      ( aa_set_int_o(finite_finite_int,insert_int(A2,A))
    <=> aa_set_int_o(finite_finite_int,A) ) ).

% finite_insert
tff(fact_7927_finite__insert,axiom,
    ! [A2: code_integer,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,insert_Code_integer(A2,A))
    <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A) ) ).

% finite_insert
tff(fact_7928_finite__insert,axiom,
    ! [A2: complex,A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,insert_complex(A2,A))
    <=> aa_set_complex_o(finite3207457112153483333omplex,A) ) ).

% finite_insert
tff(fact_7929_finite__Diff2,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(finite_finite_int,B)
     => ( aa_set_int_o(finite_finite_int,minus_minus_set_int(A,B))
      <=> aa_set_int_o(finite_finite_int,A) ) ) ).

% finite_Diff2
tff(fact_7930_finite__Diff2,axiom,
    ! [B: set_Code_integer,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A,B))
      <=> aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A) ) ) ).

% finite_Diff2
tff(fact_7931_finite__Diff2,axiom,
    ! [B: set_complex,A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => ( aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A,B))
      <=> aa_set_complex_o(finite3207457112153483333omplex,A) ) ) ).

% finite_Diff2
tff(fact_7932_finite__Diff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => ( aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A,B))
      <=> aa_set_nat_o(finite_finite_nat,A) ) ) ).

% finite_Diff2
tff(fact_7933_finite__Diff,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => aa_set_int_o(finite_finite_int,minus_minus_set_int(A,B)) ) ).

% finite_Diff
tff(fact_7934_finite__Diff,axiom,
    ! [A: set_Code_integer,B: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A,B)) ) ).

% finite_Diff
tff(fact_7935_finite__Diff,axiom,
    ! [A: set_complex,B: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A,B)) ) ).

% finite_Diff
tff(fact_7936_finite__Diff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A,B)) ) ).

% finite_Diff
tff(fact_7937_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_fz(A2),B2))) ).

% finite_interval_int3
tff(fact_7938_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_ga(A2),B2))) ).

% finite_interval_int2
tff(fact_7939_subset__Compl__singleton,axiom,
    ! [A: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A),uminus8041839845116263051T_VEBT(insert_VEBT_VEBT(B2,bot_bo8194388402131092736T_VEBT)))
    <=> ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(B2),A) ) ).

% subset_Compl_singleton
tff(fact_7940_subset__Compl__singleton,axiom,
    ! [A: set_complex,B2: complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),uminus8566677241136511917omplex(insert_complex(B2,bot_bot_set_complex)))
    <=> ~ aa_set_complex_o(member_complex(B2),A) ) ).

% subset_Compl_singleton
tff(fact_7941_subset__Compl__singleton,axiom,
    ! [A: set_real,B2: real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),uminus612125837232591019t_real(insert_real(B2,bot_bot_set_real)))
    <=> ~ aa_set_real_o(member_real(B2),A) ) ).

% subset_Compl_singleton
tff(fact_7942_subset__Compl__singleton,axiom,
    ! [A: set_o,B2: $o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),uminus_uminus_set_o(insert_o((B2),bot_bot_set_o)))
    <=> ~ aa_set_o_o(member_o((B2)),A) ) ).

% subset_Compl_singleton
tff(fact_7943_subset__Compl__singleton,axiom,
    ! [A: set_int,B2: int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),uminus1532241313380277803et_int(insert_int(B2,bot_bot_set_int)))
    <=> ~ aa_set_int_o(member_int(B2),A) ) ).

% subset_Compl_singleton
tff(fact_7944_subset__Compl__singleton,axiom,
    ! [A: set_nat,B2: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),uminus5710092332889474511et_nat(insert_nat(B2,bot_bot_set_nat)))
    <=> ~ aa_set_nat_o(member_nat(B2),A) ) ).

% subset_Compl_singleton
tff(fact_7945_negative__eq__positive,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,M) )
    <=> ( ( Nb = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
tff(fact_7946_negative__zless,axiom,
    ! [Nb: nat,M: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),aa_nat_int(semiri1314217659103216013at_int,M)) ).

% negative_zless
tff(fact_7947_nat__neg__numeral,axiom,
    ! [K: num] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = zero_zero_nat ).

% nat_neg_numeral
tff(fact_7948_nat__zminus__int,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))) = zero_zero_nat ).

% nat_zminus_int
tff(fact_7949_int__div__minus__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) = aa_int_int(uminus_uminus_int,A2) )
      <=> ( B2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_7950_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A2)),aa_num_real(numeral_numeral_real,B2))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A2))),aa_num_int(numeral_numeral_int,B2))) ).

% ceiling_divide_eq_div_numeral
tff(fact_7951_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A2)),aa_num_real(numeral_numeral_real,B2)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A2)),aa_num_int(numeral_numeral_int,B2))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_7952_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_7953_subset__Compl__self__eq,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(ord_less_eq_set_real(A),uminus612125837232591019t_real(A))
    <=> ( A = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
tff(fact_7954_subset__Compl__self__eq,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(ord_less_eq_set_o(A),uminus_uminus_set_o(A))
    <=> ( A = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
tff(fact_7955_subset__Compl__self__eq,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),uminus1532241313380277803et_int(A))
    <=> ( A = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
tff(fact_7956_subset__Compl__self__eq,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),uminus5710092332889474511et_nat(A))
    <=> ( A = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
tff(fact_7957_finite__maxlen,axiom,
    ! [M3: set_list_VEBT_VEBT] :
      ( finite3004134309566078307T_VEBT(M3)
     => ? [N: nat] :
        ! [X4: list_VEBT_VEBT] :
          ( member2936631157270082147T_VEBT(X4,M3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(X4)),N) ) ) ).

% finite_maxlen
tff(fact_7958_finite__maxlen,axiom,
    ! [M3: set_list_real] :
      ( finite306553202115118035t_real(M3)
     => ? [N: nat] :
        ! [X4: list_real] :
          ( member_list_real(X4,M3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_real(X4)),N) ) ) ).

% finite_maxlen
tff(fact_7959_finite__maxlen,axiom,
    ! [M3: set_list_o] :
      ( finite_finite_list_o(M3)
     => ? [N: nat] :
        ! [X4: list_o] :
          ( member_list_o(X4,M3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(X4)),N) ) ) ).

% finite_maxlen
tff(fact_7960_finite__maxlen,axiom,
    ! [M3: set_list_nat] :
      ( finite8100373058378681591st_nat(M3)
     => ? [N: nat] :
        ! [X4: list_nat] :
          ( member_list_nat(X4,M3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X4)),N) ) ) ).

% finite_maxlen
tff(fact_7961_finite__maxlen,axiom,
    ! [M3: set_list_VEBT_VEBTi] :
      ( finite5722435864697464412_VEBTi(M3)
     => ? [N: nat] :
        ! [X4: list_VEBT_VEBTi] :
          ( member8117914210271334748_VEBTi(X4,M3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s7982070591426661849_VEBTi(X4)),N) ) ) ).

% finite_maxlen
tff(fact_7962_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] : Z != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% int_cases
tff(fact_7963_int__of__nat__induct,axiom,
    ! [P: fun_int_o,Z: int] :
      ( ! [N: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N))
     => ( ! [N: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))))
       => aa_int_o(P,Z) ) ) ).

% int_of_nat_induct
tff(fact_7964_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(minus_minus_int(zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_7965_not__int__zless__negative,axiom,
    ! [Nb: nat,M: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M))) ).

% not_int_zless_negative
tff(fact_7966_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa_set_int_o(finite_finite_int,set_or4662586982721622107an_int(zero_zero_int,U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_7967_Compl__insert,axiom,
    ! [Xc: vEBT_VEBT,A: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(insert_VEBT_VEBT(Xc,A)) = minus_5127226145743854075T_VEBT(uminus8041839845116263051T_VEBT(A),insert_VEBT_VEBT(Xc,bot_bo8194388402131092736T_VEBT)) ).

% Compl_insert
tff(fact_7968_Compl__insert,axiom,
    ! [Xc: real,A: set_real] : uminus612125837232591019t_real(insert_real(Xc,A)) = minus_minus_set_real(uminus612125837232591019t_real(A),insert_real(Xc,bot_bot_set_real)) ).

% Compl_insert
tff(fact_7969_Compl__insert,axiom,
    ! [Xc: $o,A: set_o] : uminus_uminus_set_o(insert_o((Xc),A)) = minus_minus_set_o(uminus_uminus_set_o(A),insert_o((Xc),bot_bot_set_o)) ).

% Compl_insert
tff(fact_7970_Compl__insert,axiom,
    ! [Xc: int,A: set_int] : uminus1532241313380277803et_int(insert_int(Xc,A)) = minus_minus_set_int(uminus1532241313380277803et_int(A),insert_int(Xc,bot_bot_set_int)) ).

% Compl_insert
tff(fact_7971_Compl__insert,axiom,
    ! [Xc: nat,A: set_nat] : uminus5710092332889474511et_nat(insert_nat(Xc,A)) = minus_minus_set_nat(uminus5710092332889474511et_nat(A),insert_nat(Xc,bot_bot_set_nat)) ).

% Compl_insert
tff(fact_7972_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa_nat_int(semiri1314217659103216013at_int,N)
     => ~ ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
           => ( M != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) ) ) ) ).

% int_cases4
tff(fact_7973_int__zle__neg,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,M)))
    <=> ( ( Nb = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
tff(fact_7974_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,A2)),B2) = $ite(aa_int_int(modulo_modulo_int(A2),B2) = zero_zero_int,zero_zero_int,aa_int_int(minus_minus_int(B2),aa_int_int(modulo_modulo_int(A2),B2))) ).

% zmod_zminus1_eq_if
tff(fact_7975_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa_int_int(modulo_modulo_int(A2),aa_int_int(uminus_uminus_int,B2)) = $ite(aa_int_int(modulo_modulo_int(A2),B2) = zero_zero_int,zero_zero_int,aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(A2),B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_7976_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) )
       => ~ ! [N: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% int_cases3
tff(fact_7977_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))) ).

% not_zle_0_negative
tff(fact_7978_negD,axiom,
    ! [Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),zero_zero_int)
     => ? [N: nat] : Xc = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N))) ) ).

% negD
tff(fact_7979_negative__zless__0,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),zero_zero_int) ).

% negative_zless_0
tff(fact_7980_verit__less__mono__div__int2,axiom,
    ! [A: int,B: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,Nb))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_7981_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B2) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_7982_ceiling__divide__eq__div,axiom,
    ! [A2: int,B2: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,A2)),aa_int_real(ring_1_of_int_real,B2))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2)) ).

% ceiling_divide_eq_div
tff(fact_7983_ceiling__divide__eq__div,axiom,
    ! [A2: int,B2: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A2)),aa_int_rat(ring_1_of_int_rat,B2))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2)) ).

% ceiling_divide_eq_div
tff(fact_7984_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ).

% neg_int_cases
tff(fact_7985_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(times_times_int(Z),Z5)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_7986_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(minus_minus_int(aa_int_int(minus_minus_int(L),one_one_int)),aa_int_int(modulo_modulo_int(aa_int_int(minus_minus_int(K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_7987_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),B2) = aa_int_int(minus_minus_int(B2),one_one_int) ) ) ).

% zmod_minus1
tff(fact_7988_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_int_int(uminus_uminus_int,B2)) = $ite(aa_int_int(modulo_modulo_int(A2),B2) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_7989_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A2)),B2) = $ite(aa_int_int(modulo_modulo_int(A2),B2) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_7990_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A2,B2),product_Pair_int_int(Q3,R3))
     => ( ( B2 != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A2),B2),
            product_Pair_int_int($ite(R3 = zero_zero_int,aa_int_int(uminus_uminus_int,Q3),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,Q3)),one_one_int)),
              $ite(R3 = zero_zero_int,zero_zero_int,aa_int_int(minus_minus_int(B2),R3)))) ) ) ).

% zminus1_lemma
tff(fact_7991_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_7992_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_7993_finite__has__minimal2,axiom,
    ! [A: set_real,A2: real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_real_o(member_real(A2),A)
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),A2)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7994_finite__has__minimal2,axiom,
    ! [A: set_Code_integer,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A2),A)
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),A2)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7995_finite__has__minimal2,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( member_set_nat(A2,A)
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A)
            & aa_set_nat_o(ord_less_eq_set_nat(X3),A2)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A)
               => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7996_finite__has__minimal2,axiom,
    ! [A: set_rat,A2: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( aa_set_rat_o(member_rat(A2),A)
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),A2)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7997_finite__has__minimal2,axiom,
    ! [A: set_num,A2: num] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( aa_set_num_o(member_num(A2),A)
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),A2)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7998_finite__has__minimal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(A2),A)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),A2)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_7999_finite__has__minimal2,axiom,
    ! [A: set_int,A2: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(A2),A)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),A2)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_8000_finite__has__maximal2,axiom,
    ! [A: set_real,A2: real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_real_o(member_real(A2),A)
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),A)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X3)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8001_finite__has__maximal2,axiom,
    ! [A: set_Code_integer,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(member_Code_integer(A2),A)
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
            & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),X3)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8002_finite__has__maximal2,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( member_set_nat(A2,A)
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A)
            & aa_set_nat_o(ord_less_eq_set_nat(A2),X3)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A)
               => ( aa_set_nat_o(ord_less_eq_set_nat(X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8003_finite__has__maximal2,axiom,
    ! [A: set_rat,A2: rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( aa_set_rat_o(member_rat(A2),A)
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),X3)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8004_finite__has__maximal2,axiom,
    ! [A: set_num,A2: num] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( aa_set_num_o(member_num(A2),A)
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A)
            & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A2),X3)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8005_finite__has__maximal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(member_nat(A2),A)
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),X3)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8006_finite__has__maximal2,axiom,
    ! [A: set_int,A2: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(member_int(A2),A)
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
            & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),X3)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_8007_finite_OemptyI,axiom,
    aa_set5643387332196575111eger_o(finite6017078050557962740nteger,bot_bo3990330152332043303nteger) ).

% finite.emptyI
tff(fact_8008_finite_OemptyI,axiom,
    aa_set_complex_o(finite3207457112153483333omplex,bot_bot_set_complex) ).

% finite.emptyI
tff(fact_8009_finite_OemptyI,axiom,
    aa_set_real_o(finite_finite_real,bot_bot_set_real) ).

% finite.emptyI
tff(fact_8010_finite_OemptyI,axiom,
    aa_set_o_o(finite_finite_o,bot_bot_set_o) ).

% finite.emptyI
tff(fact_8011_finite_OemptyI,axiom,
    aa_set_nat_o(finite_finite_nat,bot_bot_set_nat) ).

% finite.emptyI
tff(fact_8012_finite_OemptyI,axiom,
    aa_set_int_o(finite_finite_int,bot_bot_set_int) ).

% finite.emptyI
tff(fact_8013_infinite__imp__nonempty,axiom,
    ! [S: set_Code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( S != bot_bo3990330152332043303nteger ) ) ).

% infinite_imp_nonempty
tff(fact_8014_infinite__imp__nonempty,axiom,
    ! [S: set_complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( S != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
tff(fact_8015_infinite__imp__nonempty,axiom,
    ! [S: set_real] :
      ( ~ aa_set_real_o(finite_finite_real,S)
     => ( S != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
tff(fact_8016_infinite__imp__nonempty,axiom,
    ! [S: set_o] :
      ( ~ aa_set_o_o(finite_finite_o,S)
     => ( S != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
tff(fact_8017_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
tff(fact_8018_infinite__imp__nonempty,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
     => ( S != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
tff(fact_8019_rev__finite__subset,axiom,
    ! [B: set_int,A: set_int] :
      ( aa_set_int_o(finite_finite_int,B)
     => ( aa_set_int_o(ord_less_eq_set_int(A),B)
       => aa_set_int_o(finite_finite_int,A) ) ) ).

% rev_finite_subset
tff(fact_8020_rev__finite__subset,axiom,
    ! [B: set_Code_integer,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),B)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A) ) ) ).

% rev_finite_subset
tff(fact_8021_rev__finite__subset,axiom,
    ! [B: set_complex,A: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(A),B)
       => aa_set_complex_o(finite3207457112153483333omplex,A) ) ) ).

% rev_finite_subset
tff(fact_8022_rev__finite__subset,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
       => aa_set_nat_o(finite_finite_nat,A) ) ) ).

% rev_finite_subset
tff(fact_8023_infinite__super,axiom,
    ! [S: set_int,T3: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(S),T3)
     => ( ~ aa_set_int_o(finite_finite_int,S)
       => ~ aa_set_int_o(finite_finite_int,T3) ) ) ).

% infinite_super
tff(fact_8024_infinite__super,axiom,
    ! [S: set_Code_integer,T3: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(S),T3)
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T3) ) ) ).

% infinite_super
tff(fact_8025_infinite__super,axiom,
    ! [S: set_complex,T3: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(S),T3)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,T3) ) ) ).

% infinite_super
tff(fact_8026_infinite__super,axiom,
    ! [S: set_nat,T3: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(S),T3)
     => ( ~ aa_set_nat_o(finite_finite_nat,S)
       => ~ aa_set_nat_o(finite_finite_nat,T3) ) ) ).

% infinite_super
tff(fact_8027_finite__subset,axiom,
    ! [A: set_int,B: set_int] :
      ( aa_set_int_o(ord_less_eq_set_int(A),B)
     => ( aa_set_int_o(finite_finite_int,B)
       => aa_set_int_o(finite_finite_int,A) ) ) ).

% finite_subset
tff(fact_8028_finite__subset,axiom,
    ! [A: set_Code_integer,B: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A),B)
     => ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B)
       => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A) ) ) ).

% finite_subset
tff(fact_8029_finite__subset,axiom,
    ! [A: set_complex,B: set_complex] :
      ( aa_set_complex_o(ord_le211207098394363844omplex(A),B)
     => ( aa_set_complex_o(finite3207457112153483333omplex,B)
       => aa_set_complex_o(finite3207457112153483333omplex,A) ) ) ).

% finite_subset
tff(fact_8030_finite__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),B)
     => ( aa_set_nat_o(finite_finite_nat,B)
       => aa_set_nat_o(finite_finite_nat,A) ) ) ).

% finite_subset
tff(fact_8031_finite_OinsertI,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,insert_VEBT_VEBT(A2,A)) ) ).

% finite.insertI
tff(fact_8032_finite_OinsertI,axiom,
    ! [A: set_real,A2: real] :
      ( aa_set_real_o(finite_finite_real,A)
     => aa_set_real_o(finite_finite_real,insert_real(A2,A)) ) ).

% finite.insertI
tff(fact_8033_finite_OinsertI,axiom,
    ! [A: set_o,A2: $o] :
      ( aa_set_o_o(finite_finite_o,A)
     => aa_set_o_o(finite_finite_o,insert_o((A2),A)) ) ).

% finite.insertI
tff(fact_8034_finite_OinsertI,axiom,
    ! [A: set_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => aa_set_nat_o(finite_finite_nat,insert_nat(A2,A)) ) ).

% finite.insertI
tff(fact_8035_finite_OinsertI,axiom,
    ! [A: set_int,A2: int] :
      ( aa_set_int_o(finite_finite_int,A)
     => aa_set_int_o(finite_finite_int,insert_int(A2,A)) ) ).

% finite.insertI
tff(fact_8036_finite_OinsertI,axiom,
    ! [A: set_Code_integer,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,insert_Code_integer(A2,A)) ) ).

% finite.insertI
tff(fact_8037_finite_OinsertI,axiom,
    ! [A: set_complex,A2: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => aa_set_complex_o(finite3207457112153483333omplex,insert_complex(A2,A)) ) ).

% finite.insertI
tff(fact_8038_Diff__infinite__finite,axiom,
    ! [T3: set_int,S: set_int] :
      ( aa_set_int_o(finite_finite_int,T3)
     => ( ~ aa_set_int_o(finite_finite_int,S)
       => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S,T3)) ) ) ).

% Diff_infinite_finite
tff(fact_8039_Diff__infinite__finite,axiom,
    ! [T3: set_Code_integer,S: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,T3)
     => ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(S,T3)) ) ) ).

% Diff_infinite_finite
tff(fact_8040_Diff__infinite__finite,axiom,
    ! [T3: set_complex,S: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,T3)
     => ( ~ aa_set_complex_o(finite3207457112153483333omplex,S)
       => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S,T3)) ) ) ).

% Diff_infinite_finite
tff(fact_8041_Diff__infinite__finite,axiom,
    ! [T3: set_nat,S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,T3)
     => ( ~ aa_set_nat_o(finite_finite_nat,S)
       => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S,T3)) ) ) ).

% Diff_infinite_finite
tff(fact_8042_finite__psubset__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ! [A7: set_nat] :
            ( aa_set_nat_o(finite_finite_nat,A7)
           => ( ! [B8: set_nat] :
                  ( aa_set_nat_o(ord_less_set_nat(B8),A7)
                 => aa_set_nat_o(P,B8) )
             => aa_set_nat_o(P,A7) ) )
       => aa_set_nat_o(P,A) ) ) ).

% finite_psubset_induct
tff(fact_8043_finite__psubset__induct,axiom,
    ! [A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ! [A7: set_int] :
            ( aa_set_int_o(finite_finite_int,A7)
           => ( ! [B8: set_int] :
                  ( aa_set_int_o(ord_less_set_int(B8),A7)
                 => aa_set_int_o(P,B8) )
             => aa_set_int_o(P,A7) ) )
       => aa_set_int_o(P,A) ) ) ).

% finite_psubset_induct
tff(fact_8044_finite__psubset__induct,axiom,
    ! [A: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ! [A7: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
           => ( ! [B8: set_Code_integer] :
                  ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(B8),A7)
                 => aa_set5643387332196575111eger_o(P,B8) )
             => aa_set5643387332196575111eger_o(P,A7) ) )
       => aa_set5643387332196575111eger_o(P,A) ) ) ).

% finite_psubset_induct
tff(fact_8045_finite__psubset__induct,axiom,
    ! [A: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( ! [A7: set_complex] :
            ( aa_set_complex_o(finite3207457112153483333omplex,A7)
           => ( ! [B8: set_complex] :
                  ( aa_set_complex_o(ord_less_set_complex(B8),A7)
                 => aa_set_complex_o(P,B8) )
             => aa_set_complex_o(P,A7) ) )
       => aa_set_complex_o(P,A) ) ) ).

% finite_psubset_induct
tff(fact_8046_int__bit__induct,axiom,
    ! [P: fun_int_o,K: int] :
      ( aa_int_o(P,zero_zero_int)
     => ( aa_int_o(P,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K2: int] :
              ( aa_int_o(P,K2)
             => ( ( K2 != zero_zero_int )
               => aa_int_o(P,aa_int_int(times_times_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ) )
         => ( ! [K2: int] :
                ( aa_int_o(P,K2)
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(P,aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_8047_m1mod2k,axiom,
    ! [Nb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) ).

% m1mod2k
tff(fact_8048_sb__dec__lem_H,axiom,
    ! [K: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)),A2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A2)) ) ).

% sb_dec_lem'
tff(fact_8049_m1mod22k,axiom,
    ! [Nb: nat] : aa_int_int(modulo_modulo_int(aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) = aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),one_one_int) ).

% m1mod22k
tff(fact_8050_sb__inc__lem_H,axiom,
    ! [A2: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))),aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,K)))) ) ).

% sb_inc_lem'
tff(fact_8051_sb__dec__lem,axiom,
    ! [K: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(modulo_modulo_int(aa_int_int(plus_plus_int(A2),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)))),aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),A2)) ) ).

% sb_dec_lem
tff(fact_8052_finite__has__minimal,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8053_finite__has__minimal,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ( A != bot_bot_set_real )
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),A)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8054_finite__has__minimal,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
            & ! [Xa: $o] :
                ( aa_set_o_o(member_o((Xa)),A)
               => ( aa_o_o(ord_less_eq_o((Xa)),(X3))
                 => ( (X3)
                  <=> (Xa) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8055_finite__has__minimal,axiom,
    ! [A: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A)
               => ( aa_set_nat_o(ord_less_eq_set_nat(Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8056_finite__has__minimal,axiom,
    ! [A: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8057_finite__has__minimal,axiom,
    ! [A: set_num] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( ( A != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8058_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8059_finite__has__minimal,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xa),X3)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_8060_finite__has__maximal,axiom,
    ! [A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( ( A != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),A)
            & ! [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),A)
               => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8061_finite__has__maximal,axiom,
    ! [A: set_real] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( ( A != bot_bot_set_real )
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),A)
            & ! [Xa: real] :
                ( aa_set_real_o(member_real(Xa),A)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8062_finite__has__maximal,axiom,
    ! [A: set_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( ( A != bot_bot_set_o )
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),A)
            & ! [Xa: $o] :
                ( aa_set_o_o(member_o((Xa)),A)
               => ( aa_o_o(ord_less_eq_o((X3)),(Xa))
                 => ( (X3)
                  <=> (Xa) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8063_finite__has__maximal,axiom,
    ! [A: set_set_nat] :
      ( finite1152437895449049373et_nat(A)
     => ( ( A != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( member_set_nat(X3,A)
            & ! [Xa: set_nat] :
                ( member_set_nat(Xa,A)
               => ( aa_set_nat_o(ord_less_eq_set_nat(X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8064_finite__has__maximal,axiom,
    ! [A: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( ( A != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),A)
            & ! [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),A)
               => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8065_finite__has__maximal,axiom,
    ! [A: set_num] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( ( A != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),A)
            & ! [Xa: num] :
                ( aa_set_num_o(member_num(Xa),A)
               => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8066_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ( A != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),A)
            & ! [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),A)
               => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8067_finite__has__maximal,axiom,
    ! [A: set_int] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( ( A != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),A)
            & ! [Xa: int] :
                ( aa_set_int_o(member_int(Xa),A)
               => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Xa)
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_8068_finite_Ocases,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ~ ! [A7: set_VEBT_VEBT] :
              ( ? [A3: vEBT_VEBT] : A2 = insert_VEBT_VEBT(A3,A7)
             => ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A7) ) ) ) ).

% finite.cases
tff(fact_8069_finite_Ocases,axiom,
    ! [A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ~ ! [A7: set_Code_integer] :
              ( ? [A3: code_integer] : A2 = insert_Code_integer(A3,A7)
             => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7) ) ) ) ).

% finite.cases
tff(fact_8070_finite_Ocases,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
     => ( ( A2 != bot_bot_set_complex )
       => ~ ! [A7: set_complex] :
              ( ? [A3: complex] : A2 = insert_complex(A3,A7)
             => ~ aa_set_complex_o(finite3207457112153483333omplex,A7) ) ) ) ).

% finite.cases
tff(fact_8071_finite_Ocases,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
     => ( ( A2 != bot_bot_set_real )
       => ~ ! [A7: set_real] :
              ( ? [A3: real] : A2 = insert_real(A3,A7)
             => ~ aa_set_real_o(finite_finite_real,A7) ) ) ) ).

% finite.cases
tff(fact_8072_finite_Ocases,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
     => ( ( A2 != bot_bot_set_o )
       => ~ ! [A7: set_o] :
              ( ? [A3: $o] : A2 = insert_o((A3),A7)
             => ~ aa_set_o_o(finite_finite_o,A7) ) ) ) ).

% finite.cases
tff(fact_8073_finite_Ocases,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
     => ( ( A2 != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A3: nat] : A2 = insert_nat(A3,A7)
             => ~ aa_set_nat_o(finite_finite_nat,A7) ) ) ) ).

% finite.cases
tff(fact_8074_finite_Ocases,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
     => ( ( A2 != bot_bot_set_int )
       => ~ ! [A7: set_int] :
              ( ? [A3: int] : A2 = insert_int(A3,A7)
             => ~ aa_set_int_o(finite_finite_int,A7) ) ) ) ).

% finite.cases
tff(fact_8075_finite_Osimps,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A2)
    <=> ( ( A2 = bot_bo8194388402131092736T_VEBT )
        | ? [A8: set_VEBT_VEBT,A6: vEBT_VEBT] :
            ( ( A2 = insert_VEBT_VEBT(A6,A8) )
            & aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A8) ) ) ) ).

% finite.simps
tff(fact_8076_finite_Osimps,axiom,
    ! [A2: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A2)
    <=> ( ( A2 = bot_bo3990330152332043303nteger )
        | ? [A8: set_Code_integer,A6: code_integer] :
            ( ( A2 = insert_Code_integer(A6,A8) )
            & aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A8) ) ) ) ).

% finite.simps
tff(fact_8077_finite_Osimps,axiom,
    ! [A2: set_complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A2)
    <=> ( ( A2 = bot_bot_set_complex )
        | ? [A8: set_complex,A6: complex] :
            ( ( A2 = insert_complex(A6,A8) )
            & aa_set_complex_o(finite3207457112153483333omplex,A8) ) ) ) ).

% finite.simps
tff(fact_8078_finite_Osimps,axiom,
    ! [A2: set_real] :
      ( aa_set_real_o(finite_finite_real,A2)
    <=> ( ( A2 = bot_bot_set_real )
        | ? [A8: set_real,A6: real] :
            ( ( A2 = insert_real(A6,A8) )
            & aa_set_real_o(finite_finite_real,A8) ) ) ) ).

% finite.simps
tff(fact_8079_finite_Osimps,axiom,
    ! [A2: set_o] :
      ( aa_set_o_o(finite_finite_o,A2)
    <=> ( ( A2 = bot_bot_set_o )
        | ? [A8: set_o,A6: $o] :
            ( ( A2 = insert_o((A6),A8) )
            & aa_set_o_o(finite_finite_o,A8) ) ) ) ).

% finite.simps
tff(fact_8080_finite_Osimps,axiom,
    ! [A2: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A2)
    <=> ( ( A2 = bot_bot_set_nat )
        | ? [A8: set_nat,A6: nat] :
            ( ( A2 = insert_nat(A6,A8) )
            & aa_set_nat_o(finite_finite_nat,A8) ) ) ) ).

% finite.simps
tff(fact_8081_finite_Osimps,axiom,
    ! [A2: set_int] :
      ( aa_set_int_o(finite_finite_int,A2)
    <=> ( ( A2 = bot_bot_set_int )
        | ? [A8: set_int,A6: int] :
            ( ( A2 = insert_int(A6,A8) )
            & aa_set_int_o(finite_finite_int,A8) ) ) ) ).

% finite.simps
tff(fact_8082_finite__induct,axiom,
    ! [F: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X3: vEBT_VEBT,F6: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F6)
             => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),F6)
               => ( aa_set_VEBT_VEBT_o(P,F6)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X3,F6)) ) ) )
         => aa_set_VEBT_VEBT_o(P,F) ) ) ) ).

% finite_induct
tff(fact_8083_finite__induct,axiom,
    ! [F: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X3: code_integer,F6: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F6)
             => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X3),F6)
               => ( aa_set5643387332196575111eger_o(P,F6)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(X3,F6)) ) ) )
         => aa_set5643387332196575111eger_o(P,F) ) ) ) ).

% finite_induct
tff(fact_8084_finite__induct,axiom,
    ! [F: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,F6: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F6)
             => ( ~ aa_set_complex_o(member_complex(X3),F6)
               => ( aa_set_complex_o(P,F6)
                 => aa_set_complex_o(P,insert_complex(X3,F6)) ) ) )
         => aa_set_complex_o(P,F) ) ) ) ).

% finite_induct
tff(fact_8085_finite__induct,axiom,
    ! [F: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,F6: set_real] :
              ( aa_set_real_o(finite_finite_real,F6)
             => ( ~ aa_set_real_o(member_real(X3),F6)
               => ( aa_set_real_o(P,F6)
                 => aa_set_real_o(P,insert_real(X3,F6)) ) ) )
         => aa_set_real_o(P,F) ) ) ) ).

% finite_induct
tff(fact_8086_finite__induct,axiom,
    ! [F: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X3: $o,F6: set_o] :
              ( aa_set_o_o(finite_finite_o,F6)
             => ( ~ aa_set_o_o(member_o((X3)),F6)
               => ( aa_set_o_o(P,F6)
                 => aa_set_o_o(P,insert_o((X3),F6)) ) ) )
         => aa_set_o_o(P,F) ) ) ) ).

% finite_induct
tff(fact_8087_finite__induct,axiom,
    ! [F: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,F6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F6)
             => ( ~ aa_set_nat_o(member_nat(X3),F6)
               => ( aa_set_nat_o(P,F6)
                 => aa_set_nat_o(P,insert_nat(X3,F6)) ) ) )
         => aa_set_nat_o(P,F) ) ) ) ).

% finite_induct
tff(fact_8088_finite__induct,axiom,
    ! [F: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,F6: set_int] :
              ( aa_set_int_o(finite_finite_int,F6)
             => ( ~ aa_set_int_o(member_int(X3),F6)
               => ( aa_set_int_o(P,F6)
                 => aa_set_int_o(P,insert_int(X3,F6)) ) ) )
         => aa_set_int_o(P,F) ) ) ) ).

% finite_induct
tff(fact_8089_finite__ne__induct,axiom,
    ! [F: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F)
     => ( ( F != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] : aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X3,bot_bo8194388402131092736T_VEBT))
         => ( ! [X3: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F6)
               => ( ( F6 != bot_bo8194388402131092736T_VEBT )
                 => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),F6)
                   => ( aa_set_VEBT_VEBT_o(P,F6)
                     => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X3,F6)) ) ) ) )
           => aa_set_VEBT_VEBT_o(P,F) ) ) ) ) ).

% finite_ne_induct
tff(fact_8090_finite__ne__induct,axiom,
    ! [F: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F)
     => ( ( F != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] : aa_set5643387332196575111eger_o(P,insert_Code_integer(X3,bot_bo3990330152332043303nteger))
         => ( ! [X3: code_integer,F6: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F6)
               => ( ( F6 != bot_bo3990330152332043303nteger )
                 => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X3),F6)
                   => ( aa_set5643387332196575111eger_o(P,F6)
                     => aa_set5643387332196575111eger_o(P,insert_Code_integer(X3,F6)) ) ) ) )
           => aa_set5643387332196575111eger_o(P,F) ) ) ) ) ).

% finite_ne_induct
tff(fact_8091_finite__ne__induct,axiom,
    ! [F: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F)
     => ( ( F != bot_bot_set_complex )
       => ( ! [X3: complex] : aa_set_complex_o(P,insert_complex(X3,bot_bot_set_complex))
         => ( ! [X3: complex,F6: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F6)
               => ( ( F6 != bot_bot_set_complex )
                 => ( ~ aa_set_complex_o(member_complex(X3),F6)
                   => ( aa_set_complex_o(P,F6)
                     => aa_set_complex_o(P,insert_complex(X3,F6)) ) ) ) )
           => aa_set_complex_o(P,F) ) ) ) ) ).

% finite_ne_induct
tff(fact_8092_finite__ne__induct,axiom,
    ! [F: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F)
     => ( ( F != bot_bot_set_real )
       => ( ! [X3: real] : aa_set_real_o(P,insert_real(X3,bot_bot_set_real))
         => ( ! [X3: real,F6: set_real] :
                ( aa_set_real_o(finite_finite_real,F6)
               => ( ( F6 != bot_bot_set_real )
                 => ( ~ aa_set_real_o(member_real(X3),F6)
                   => ( aa_set_real_o(P,F6)
                     => aa_set_real_o(P,insert_real(X3,F6)) ) ) ) )
           => aa_set_real_o(P,F) ) ) ) ) ).

% finite_ne_induct
tff(fact_8093_finite__ne__induct,axiom,
    ! [F: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F)
     => ( ( F != bot_bot_set_o )
       => ( ! [X3: $o] : aa_set_o_o(P,insert_o((X3),bot_bot_set_o))
         => ( ! [X3: $o,F6: set_o] :
                ( aa_set_o_o(finite_finite_o,F6)
               => ( ( F6 != bot_bot_set_o )
                 => ( ~ aa_set_o_o(member_o((X3)),F6)
                   => ( aa_set_o_o(P,F6)
                     => aa_set_o_o(P,insert_o((X3),F6)) ) ) ) )
           => aa_set_o_o(P,F) ) ) ) ) ).

% finite_ne_induct
tff(fact_8094_finite__ne__induct,axiom,
    ! [F: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F)
     => ( ( F != bot_bot_set_nat )
       => ( ! [X3: nat] : aa_set_nat_o(P,insert_nat(X3,bot_bot_set_nat))
         => ( ! [X3: nat,F6: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F6)
               => ( ( F6 != bot_bot_set_nat )
                 => ( ~ aa_set_nat_o(member_nat(X3),F6)
                   => ( aa_set_nat_o(P,F6)
                     => aa_set_nat_o(P,insert_nat(X3,F6)) ) ) ) )
           => aa_set_nat_o(P,F) ) ) ) ) ).

% finite_ne_induct
tff(fact_8095_finite__ne__induct,axiom,
    ! [F: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F)
     => ( ( F != bot_bot_set_int )
       => ( ! [X3: int] : aa_set_int_o(P,insert_int(X3,bot_bot_set_int))
         => ( ! [X3: int,F6: set_int] :
                ( aa_set_int_o(finite_finite_int,F6)
               => ( ( F6 != bot_bot_set_int )
                 => ( ~ aa_set_int_o(member_int(X3),F6)
                   => ( aa_set_int_o(P,F6)
                     => aa_set_int_o(P,insert_int(X3,F6)) ) ) ) )
           => aa_set_int_o(P,F) ) ) ) ) ).

% finite_ne_induct
tff(fact_8096_infinite__finite__induct,axiom,
    ! [P: fun_set_VEBT_VEBT_o,A: set_VEBT_VEBT] :
      ( ! [A7: set_VEBT_VEBT] :
          ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A7)
         => aa_set_VEBT_VEBT_o(P,A7) )
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X3: vEBT_VEBT,F6: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F6)
             => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),F6)
               => ( aa_set_VEBT_VEBT_o(P,F6)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X3,F6)) ) ) )
         => aa_set_VEBT_VEBT_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_8097_infinite__finite__induct,axiom,
    ! [P: fun_se7289222521248229150eger_o,A: set_Code_integer] :
      ( ! [A7: set_Code_integer] :
          ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
         => aa_set5643387332196575111eger_o(P,A7) )
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X3: code_integer,F6: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F6)
             => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(X3),F6)
               => ( aa_set5643387332196575111eger_o(P,F6)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(X3,F6)) ) ) )
         => aa_set5643387332196575111eger_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_8098_infinite__finite__induct,axiom,
    ! [P: fun_set_complex_o,A: set_complex] :
      ( ! [A7: set_complex] :
          ( ~ aa_set_complex_o(finite3207457112153483333omplex,A7)
         => aa_set_complex_o(P,A7) )
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,F6: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,F6)
             => ( ~ aa_set_complex_o(member_complex(X3),F6)
               => ( aa_set_complex_o(P,F6)
                 => aa_set_complex_o(P,insert_complex(X3,F6)) ) ) )
         => aa_set_complex_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_8099_infinite__finite__induct,axiom,
    ! [P: fun_set_real_o,A: set_real] :
      ( ! [A7: set_real] :
          ( ~ aa_set_real_o(finite_finite_real,A7)
         => aa_set_real_o(P,A7) )
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,F6: set_real] :
              ( aa_set_real_o(finite_finite_real,F6)
             => ( ~ aa_set_real_o(member_real(X3),F6)
               => ( aa_set_real_o(P,F6)
                 => aa_set_real_o(P,insert_real(X3,F6)) ) ) )
         => aa_set_real_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_8100_infinite__finite__induct,axiom,
    ! [P: fun_set_o_o,A: set_o] :
      ( ! [A7: set_o] :
          ( ~ aa_set_o_o(finite_finite_o,A7)
         => aa_set_o_o(P,A7) )
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X3: $o,F6: set_o] :
              ( aa_set_o_o(finite_finite_o,F6)
             => ( ~ aa_set_o_o(member_o((X3)),F6)
               => ( aa_set_o_o(P,F6)
                 => aa_set_o_o(P,insert_o((X3),F6)) ) ) )
         => aa_set_o_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_8101_infinite__finite__induct,axiom,
    ! [P: fun_set_nat_o,A: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ aa_set_nat_o(finite_finite_nat,A7)
         => aa_set_nat_o(P,A7) )
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,F6: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,F6)
             => ( ~ aa_set_nat_o(member_nat(X3),F6)
               => ( aa_set_nat_o(P,F6)
                 => aa_set_nat_o(P,insert_nat(X3,F6)) ) ) )
         => aa_set_nat_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_8102_infinite__finite__induct,axiom,
    ! [P: fun_set_int_o,A: set_int] :
      ( ! [A7: set_int] :
          ( ~ aa_set_int_o(finite_finite_int,A7)
         => aa_set_int_o(P,A7) )
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,F6: set_int] :
              ( aa_set_int_o(finite_finite_int,F6)
             => ( ~ aa_set_int_o(member_int(X3),F6)
               => ( aa_set_int_o(P,F6)
                 => aa_set_int_o(P,insert_int(X3,F6)) ) ) )
         => aa_set_int_o(P,A) ) ) ) ).

% infinite_finite_induct
tff(fact_8103_finite__subset__induct,axiom,
    ! [F: set_VEBT_VEBT,A: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F),A)
       => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
         => ( ! [A3: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F6)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A)
                 => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),F6)
                   => ( aa_set_VEBT_VEBT_o(P,F6)
                     => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(A3,F6)) ) ) ) )
           => aa_set_VEBT_VEBT_o(P,F) ) ) ) ) ).

% finite_subset_induct
tff(fact_8104_finite__subset__induct,axiom,
    ! [F: set_Code_integer,A: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F),A)
       => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
         => ( ! [A3: code_integer,F6: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F6)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A)
                 => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(A3),F6)
                   => ( aa_set5643387332196575111eger_o(P,F6)
                     => aa_set5643387332196575111eger_o(P,insert_Code_integer(A3,F6)) ) ) ) )
           => aa_set5643387332196575111eger_o(P,F) ) ) ) ) ).

% finite_subset_induct
tff(fact_8105_finite__subset__induct,axiom,
    ! [F: set_complex,A: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F),A)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A3: complex,F6: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F6)
               => ( aa_set_complex_o(member_complex(A3),A)
                 => ( ~ aa_set_complex_o(member_complex(A3),F6)
                   => ( aa_set_complex_o(P,F6)
                     => aa_set_complex_o(P,insert_complex(A3,F6)) ) ) ) )
           => aa_set_complex_o(P,F) ) ) ) ) ).

% finite_subset_induct
tff(fact_8106_finite__subset__induct,axiom,
    ! [F: set_real,A: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F)
     => ( aa_set_real_o(ord_less_eq_set_real(F),A)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A3: real,F6: set_real] :
                ( aa_set_real_o(finite_finite_real,F6)
               => ( aa_set_real_o(member_real(A3),A)
                 => ( ~ aa_set_real_o(member_real(A3),F6)
                   => ( aa_set_real_o(P,F6)
                     => aa_set_real_o(P,insert_real(A3,F6)) ) ) ) )
           => aa_set_real_o(P,F) ) ) ) ) ).

% finite_subset_induct
tff(fact_8107_finite__subset__induct,axiom,
    ! [F: set_o,A: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F)
     => ( aa_set_o_o(ord_less_eq_set_o(F),A)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A3: $o,F6: set_o] :
                ( aa_set_o_o(finite_finite_o,F6)
               => ( aa_set_o_o(member_o((A3)),A)
                 => ( ~ aa_set_o_o(member_o((A3)),F6)
                   => ( aa_set_o_o(P,F6)
                     => aa_set_o_o(P,insert_o((A3),F6)) ) ) ) )
           => aa_set_o_o(P,F) ) ) ) ) ).

% finite_subset_induct
tff(fact_8108_finite__subset__induct,axiom,
    ! [F: set_int,A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F)
     => ( aa_set_int_o(ord_less_eq_set_int(F),A)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A3: int,F6: set_int] :
                ( aa_set_int_o(finite_finite_int,F6)
               => ( aa_set_int_o(member_int(A3),A)
                 => ( ~ aa_set_int_o(member_int(A3),F6)
                   => ( aa_set_int_o(P,F6)
                     => aa_set_int_o(P,insert_int(A3,F6)) ) ) ) )
           => aa_set_int_o(P,F) ) ) ) ) ).

% finite_subset_induct
tff(fact_8109_finite__subset__induct,axiom,
    ! [F: set_nat,A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F)
     => ( aa_set_nat_o(ord_less_eq_set_nat(F),A)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A3: nat,F6: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F6)
               => ( aa_set_nat_o(member_nat(A3),A)
                 => ( ~ aa_set_nat_o(member_nat(A3),F6)
                   => ( aa_set_nat_o(P,F6)
                     => aa_set_nat_o(P,insert_nat(A3,F6)) ) ) ) )
           => aa_set_nat_o(P,F) ) ) ) ) ).

% finite_subset_induct
tff(fact_8110_finite__subset__induct_H,axiom,
    ! [F: set_VEBT_VEBT,A: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F)
     => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F),A)
       => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
         => ( ! [A3: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,F6)
               => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A)
                 => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(F6),A)
                   => ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),F6)
                     => ( aa_set_VEBT_VEBT_o(P,F6)
                       => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(A3,F6)) ) ) ) ) )
           => aa_set_VEBT_VEBT_o(P,F) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8111_finite__subset__induct_H,axiom,
    ! [F: set_Code_integer,A: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F)
     => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F),A)
       => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
         => ( ! [A3: code_integer,F6: set_Code_integer] :
                ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,F6)
               => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A)
                 => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(F6),A)
                   => ( ~ aa_set5643387332196575111eger_o(member_Code_integer(A3),F6)
                     => ( aa_set5643387332196575111eger_o(P,F6)
                       => aa_set5643387332196575111eger_o(P,insert_Code_integer(A3,F6)) ) ) ) ) )
           => aa_set5643387332196575111eger_o(P,F) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8112_finite__subset__induct_H,axiom,
    ! [F: set_complex,A: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,F)
     => ( aa_set_complex_o(ord_le211207098394363844omplex(F),A)
       => ( aa_set_complex_o(P,bot_bot_set_complex)
         => ( ! [A3: complex,F6: set_complex] :
                ( aa_set_complex_o(finite3207457112153483333omplex,F6)
               => ( aa_set_complex_o(member_complex(A3),A)
                 => ( aa_set_complex_o(ord_le211207098394363844omplex(F6),A)
                   => ( ~ aa_set_complex_o(member_complex(A3),F6)
                     => ( aa_set_complex_o(P,F6)
                       => aa_set_complex_o(P,insert_complex(A3,F6)) ) ) ) ) )
           => aa_set_complex_o(P,F) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8113_finite__subset__induct_H,axiom,
    ! [F: set_real,A: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,F)
     => ( aa_set_real_o(ord_less_eq_set_real(F),A)
       => ( aa_set_real_o(P,bot_bot_set_real)
         => ( ! [A3: real,F6: set_real] :
                ( aa_set_real_o(finite_finite_real,F6)
               => ( aa_set_real_o(member_real(A3),A)
                 => ( aa_set_real_o(ord_less_eq_set_real(F6),A)
                   => ( ~ aa_set_real_o(member_real(A3),F6)
                     => ( aa_set_real_o(P,F6)
                       => aa_set_real_o(P,insert_real(A3,F6)) ) ) ) ) )
           => aa_set_real_o(P,F) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8114_finite__subset__induct_H,axiom,
    ! [F: set_o,A: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,F)
     => ( aa_set_o_o(ord_less_eq_set_o(F),A)
       => ( aa_set_o_o(P,bot_bot_set_o)
         => ( ! [A3: $o,F6: set_o] :
                ( aa_set_o_o(finite_finite_o,F6)
               => ( aa_set_o_o(member_o((A3)),A)
                 => ( aa_set_o_o(ord_less_eq_set_o(F6),A)
                   => ( ~ aa_set_o_o(member_o((A3)),F6)
                     => ( aa_set_o_o(P,F6)
                       => aa_set_o_o(P,insert_o((A3),F6)) ) ) ) ) )
           => aa_set_o_o(P,F) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8115_finite__subset__induct_H,axiom,
    ! [F: set_int,A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,F)
     => ( aa_set_int_o(ord_less_eq_set_int(F),A)
       => ( aa_set_int_o(P,bot_bot_set_int)
         => ( ! [A3: int,F6: set_int] :
                ( aa_set_int_o(finite_finite_int,F6)
               => ( aa_set_int_o(member_int(A3),A)
                 => ( aa_set_int_o(ord_less_eq_set_int(F6),A)
                   => ( ~ aa_set_int_o(member_int(A3),F6)
                     => ( aa_set_int_o(P,F6)
                       => aa_set_int_o(P,insert_int(A3,F6)) ) ) ) ) )
           => aa_set_int_o(P,F) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8116_finite__subset__induct_H,axiom,
    ! [F: set_nat,A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,F)
     => ( aa_set_nat_o(ord_less_eq_set_nat(F),A)
       => ( aa_set_nat_o(P,bot_bot_set_nat)
         => ( ! [A3: nat,F6: set_nat] :
                ( aa_set_nat_o(finite_finite_nat,F6)
               => ( aa_set_nat_o(member_nat(A3),A)
                 => ( aa_set_nat_o(ord_less_eq_set_nat(F6),A)
                   => ( ~ aa_set_nat_o(member_nat(A3),F6)
                     => ( aa_set_nat_o(P,F6)
                       => aa_set_nat_o(P,insert_nat(A3,F6)) ) ) ) ) )
           => aa_set_nat_o(P,F) ) ) ) ) ).

% finite_subset_induct'
tff(fact_8117_infinite__remove,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S)
     => ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(S,insert_VEBT_VEBT(A2,bot_bo8194388402131092736T_VEBT))) ) ).

% infinite_remove
tff(fact_8118_infinite__remove,axiom,
    ! [S: set_Code_integer,A2: code_integer] :
      ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(S,insert_Code_integer(A2,bot_bo3990330152332043303nteger))) ) ).

% infinite_remove
tff(fact_8119_infinite__remove,axiom,
    ! [S: set_complex,A2: complex] :
      ( ~ aa_set_complex_o(finite3207457112153483333omplex,S)
     => ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(S,insert_complex(A2,bot_bot_set_complex))) ) ).

% infinite_remove
tff(fact_8120_infinite__remove,axiom,
    ! [S: set_real,A2: real] :
      ( ~ aa_set_real_o(finite_finite_real,S)
     => ~ aa_set_real_o(finite_finite_real,minus_minus_set_real(S,insert_real(A2,bot_bot_set_real))) ) ).

% infinite_remove
tff(fact_8121_infinite__remove,axiom,
    ! [S: set_o,A2: $o] :
      ( ~ aa_set_o_o(finite_finite_o,S)
     => ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(S,insert_o((A2),bot_bot_set_o))) ) ).

% infinite_remove
tff(fact_8122_infinite__remove,axiom,
    ! [S: set_int,A2: int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
     => ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(S,insert_int(A2,bot_bot_set_int))) ) ).

% infinite_remove
tff(fact_8123_infinite__remove,axiom,
    ! [S: set_nat,A2: nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
     => ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(S,insert_nat(A2,bot_bot_set_nat))) ) ).

% infinite_remove
tff(fact_8124_infinite__coinduct,axiom,
    ! [X: fun_set_VEBT_VEBT_o,A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(X,A)
     => ( ! [A7: set_VEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(X,A7)
           => ? [X4: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),A7)
                & ( aa_set_VEBT_VEBT_o(X,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X4,bot_bo8194388402131092736T_VEBT)))
                  | ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X4,bot_bo8194388402131092736T_VEBT))) ) ) )
       => ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A) ) ) ).

% infinite_coinduct
tff(fact_8125_infinite__coinduct,axiom,
    ! [X: fun_se7289222521248229150eger_o,A: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(X,A)
     => ( ! [A7: set_Code_integer] :
            ( aa_set5643387332196575111eger_o(X,A7)
           => ? [X4: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A7)
                & ( aa_set5643387332196575111eger_o(X,minus_2355218937544613996nteger(A7,insert_Code_integer(X4,bot_bo3990330152332043303nteger)))
                  | ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,minus_2355218937544613996nteger(A7,insert_Code_integer(X4,bot_bo3990330152332043303nteger))) ) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A) ) ) ).

% infinite_coinduct
tff(fact_8126_infinite__coinduct,axiom,
    ! [X: fun_set_complex_o,A: set_complex] :
      ( aa_set_complex_o(X,A)
     => ( ! [A7: set_complex] :
            ( aa_set_complex_o(X,A7)
           => ? [X4: complex] :
                ( aa_set_complex_o(member_complex(X4),A7)
                & ( aa_set_complex_o(X,minus_811609699411566653omplex(A7,insert_complex(X4,bot_bot_set_complex)))
                  | ~ aa_set_complex_o(finite3207457112153483333omplex,minus_811609699411566653omplex(A7,insert_complex(X4,bot_bot_set_complex))) ) ) )
       => ~ aa_set_complex_o(finite3207457112153483333omplex,A) ) ) ).

% infinite_coinduct
tff(fact_8127_infinite__coinduct,axiom,
    ! [X: fun_set_real_o,A: set_real] :
      ( aa_set_real_o(X,A)
     => ( ! [A7: set_real] :
            ( aa_set_real_o(X,A7)
           => ? [X4: real] :
                ( aa_set_real_o(member_real(X4),A7)
                & ( aa_set_real_o(X,minus_minus_set_real(A7,insert_real(X4,bot_bot_set_real)))
                  | ~ aa_set_real_o(finite_finite_real,minus_minus_set_real(A7,insert_real(X4,bot_bot_set_real))) ) ) )
       => ~ aa_set_real_o(finite_finite_real,A) ) ) ).

% infinite_coinduct
tff(fact_8128_infinite__coinduct,axiom,
    ! [X: fun_set_o_o,A: set_o] :
      ( aa_set_o_o(X,A)
     => ( ! [A7: set_o] :
            ( aa_set_o_o(X,A7)
           => ? [X4: $o] :
                ( aa_set_o_o(member_o((X4)),A7)
                & ( aa_set_o_o(X,minus_minus_set_o(A7,insert_o((X4),bot_bot_set_o)))
                  | ~ aa_set_o_o(finite_finite_o,minus_minus_set_o(A7,insert_o((X4),bot_bot_set_o))) ) ) )
       => ~ aa_set_o_o(finite_finite_o,A) ) ) ).

% infinite_coinduct
tff(fact_8129_infinite__coinduct,axiom,
    ! [X: fun_set_int_o,A: set_int] :
      ( aa_set_int_o(X,A)
     => ( ! [A7: set_int] :
            ( aa_set_int_o(X,A7)
           => ? [X4: int] :
                ( aa_set_int_o(member_int(X4),A7)
                & ( aa_set_int_o(X,minus_minus_set_int(A7,insert_int(X4,bot_bot_set_int)))
                  | ~ aa_set_int_o(finite_finite_int,minus_minus_set_int(A7,insert_int(X4,bot_bot_set_int))) ) ) )
       => ~ aa_set_int_o(finite_finite_int,A) ) ) ).

% infinite_coinduct
tff(fact_8130_infinite__coinduct,axiom,
    ! [X: fun_set_nat_o,A: set_nat] :
      ( aa_set_nat_o(X,A)
     => ( ! [A7: set_nat] :
            ( aa_set_nat_o(X,A7)
           => ? [X4: nat] :
                ( aa_set_nat_o(member_nat(X4),A7)
                & ( aa_set_nat_o(X,minus_minus_set_nat(A7,insert_nat(X4,bot_bot_set_nat)))
                  | ~ aa_set_nat_o(finite_finite_nat,minus_minus_set_nat(A7,insert_nat(X4,bot_bot_set_nat))) ) ) )
       => ~ aa_set_nat_o(finite_finite_nat,A) ) ) ).

% infinite_coinduct
tff(fact_8131_finite__empty__induct,axiom,
    ! [A: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A)
     => ( aa_set_VEBT_VEBT_o(P,A)
       => ( ! [A3: vEBT_VEBT,A7: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A7)
             => ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(A3),A7)
               => ( aa_set_VEBT_VEBT_o(P,A7)
                 => aa_set_VEBT_VEBT_o(P,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(A3,bot_bo8194388402131092736T_VEBT))) ) ) )
         => aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT) ) ) ) ).

% finite_empty_induct
tff(fact_8132_finite__empty__induct,axiom,
    ! [A: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(P,A)
       => ( ! [A3: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( aa_set5643387332196575111eger_o(member_Code_integer(A3),A7)
               => ( aa_set5643387332196575111eger_o(P,A7)
                 => aa_set5643387332196575111eger_o(P,minus_2355218937544613996nteger(A7,insert_Code_integer(A3,bot_bo3990330152332043303nteger))) ) ) )
         => aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger) ) ) ) ).

% finite_empty_induct
tff(fact_8133_finite__empty__induct,axiom,
    ! [A: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,A)
     => ( aa_set_complex_o(P,A)
       => ( ! [A3: complex,A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( aa_set_complex_o(member_complex(A3),A7)
               => ( aa_set_complex_o(P,A7)
                 => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(A3,bot_bot_set_complex))) ) ) )
         => aa_set_complex_o(P,bot_bot_set_complex) ) ) ) ).

% finite_empty_induct
tff(fact_8134_finite__empty__induct,axiom,
    ! [A: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_real_o(P,A)
       => ( ! [A3: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( aa_set_real_o(member_real(A3),A7)
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(A3,bot_bot_set_real))) ) ) )
         => aa_set_real_o(P,bot_bot_set_real) ) ) ) ).

% finite_empty_induct
tff(fact_8135_finite__empty__induct,axiom,
    ! [A: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(P,A)
       => ( ! [A3: $o,A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( aa_set_o_o(member_o((A3)),A7)
               => ( aa_set_o_o(P,A7)
                 => aa_set_o_o(P,minus_minus_set_o(A7,insert_o((A3),bot_bot_set_o))) ) ) )
         => aa_set_o_o(P,bot_bot_set_o) ) ) ) ).

% finite_empty_induct
tff(fact_8136_finite__empty__induct,axiom,
    ! [A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(P,A)
       => ( ! [A3: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( aa_set_int_o(member_int(A3),A7)
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(A3,bot_bot_set_int))) ) ) )
         => aa_set_int_o(P,bot_bot_set_int) ) ) ) ).

% finite_empty_induct
tff(fact_8137_finite__empty__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(P,A)
       => ( ! [A3: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( aa_set_nat_o(member_nat(A3),A7)
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,minus_minus_set_nat(A7,insert_nat(A3,bot_bot_set_nat))) ) ) )
         => aa_set_nat_o(P,bot_bot_set_nat) ) ) ) ).

% finite_empty_induct
tff(fact_8138_remove__induct,axiom,
    ! [P: fun_set_VEBT_VEBT_o,B: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
     => ( ( ~ aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B)
         => aa_set_VEBT_VEBT_o(P,B) )
       => ( ! [A7: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A7)
             => ( ( A7 != bot_bo8194388402131092736T_VEBT )
               => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A7),B)
                 => ( ! [X4: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),A7)
                       => aa_set_VEBT_VEBT_o(P,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X4,bot_bo8194388402131092736T_VEBT))) )
                   => aa_set_VEBT_VEBT_o(P,A7) ) ) ) )
         => aa_set_VEBT_VEBT_o(P,B) ) ) ) ).

% remove_induct
tff(fact_8139_remove__induct,axiom,
    ! [P: fun_se7289222521248229150eger_o,B: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
     => ( ( ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B)
         => aa_set5643387332196575111eger_o(P,B) )
       => ( ! [A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A7),B)
                 => ( ! [X4: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A7)
                       => aa_set5643387332196575111eger_o(P,minus_2355218937544613996nteger(A7,insert_Code_integer(X4,bot_bo3990330152332043303nteger))) )
                   => aa_set5643387332196575111eger_o(P,A7) ) ) ) )
         => aa_set5643387332196575111eger_o(P,B) ) ) ) ).

% remove_induct
tff(fact_8140_remove__induct,axiom,
    ! [P: fun_set_complex_o,B: set_complex] :
      ( aa_set_complex_o(P,bot_bot_set_complex)
     => ( ( ~ aa_set_complex_o(finite3207457112153483333omplex,B)
         => aa_set_complex_o(P,B) )
       => ( ! [A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( ( A7 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A7),B)
                 => ( ! [X4: complex] :
                        ( aa_set_complex_o(member_complex(X4),A7)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(X4,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A7) ) ) ) )
         => aa_set_complex_o(P,B) ) ) ) ).

% remove_induct
tff(fact_8141_remove__induct,axiom,
    ! [P: fun_set_real_o,B: set_real] :
      ( aa_set_real_o(P,bot_bot_set_real)
     => ( ( ~ aa_set_real_o(finite_finite_real,B)
         => aa_set_real_o(P,B) )
       => ( ! [A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ( A7 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A7),B)
                 => ( ! [X4: real] :
                        ( aa_set_real_o(member_real(X4),A7)
                       => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(X4,bot_bot_set_real))) )
                   => aa_set_real_o(P,A7) ) ) ) )
         => aa_set_real_o(P,B) ) ) ) ).

% remove_induct
tff(fact_8142_remove__induct,axiom,
    ! [P: fun_set_o_o,B: set_o] :
      ( aa_set_o_o(P,bot_bot_set_o)
     => ( ( ~ aa_set_o_o(finite_finite_o,B)
         => aa_set_o_o(P,B) )
       => ( ! [A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ( A7 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A7),B)
                 => ( ! [X4: $o] :
                        ( aa_set_o_o(member_o((X4)),A7)
                       => aa_set_o_o(P,minus_minus_set_o(A7,insert_o((X4),bot_bot_set_o))) )
                   => aa_set_o_o(P,A7) ) ) ) )
         => aa_set_o_o(P,B) ) ) ) ).

% remove_induct
tff(fact_8143_remove__induct,axiom,
    ! [P: fun_set_int_o,B: set_int] :
      ( aa_set_int_o(P,bot_bot_set_int)
     => ( ( ~ aa_set_int_o(finite_finite_int,B)
         => aa_set_int_o(P,B) )
       => ( ! [A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ( A7 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A7),B)
                 => ( ! [X4: int] :
                        ( aa_set_int_o(member_int(X4),A7)
                       => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(X4,bot_bot_set_int))) )
                   => aa_set_int_o(P,A7) ) ) ) )
         => aa_set_int_o(P,B) ) ) ) ).

% remove_induct
tff(fact_8144_remove__induct,axiom,
    ! [P: fun_set_nat_o,B: set_nat] :
      ( aa_set_nat_o(P,bot_bot_set_nat)
     => ( ( ~ aa_set_nat_o(finite_finite_nat,B)
         => aa_set_nat_o(P,B) )
       => ( ! [A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ( A7 != bot_bot_set_nat )
               => ( aa_set_nat_o(ord_less_eq_set_nat(A7),B)
                 => ( ! [X4: nat] :
                        ( aa_set_nat_o(member_nat(X4),A7)
                       => aa_set_nat_o(P,minus_minus_set_nat(A7,insert_nat(X4,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A7) ) ) ) )
         => aa_set_nat_o(P,B) ) ) ) ).

% remove_induct
tff(fact_8145_finite__remove__induct,axiom,
    ! [B: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,B)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [A7: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,A7)
             => ( ( A7 != bot_bo8194388402131092736T_VEBT )
               => ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(A7),B)
                 => ( ! [X4: vEBT_VEBT] :
                        ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),A7)
                       => aa_set_VEBT_VEBT_o(P,minus_5127226145743854075T_VEBT(A7,insert_VEBT_VEBT(X4,bot_bo8194388402131092736T_VEBT))) )
                   => aa_set_VEBT_VEBT_o(P,A7) ) ) ) )
         => aa_set_VEBT_VEBT_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_8146_finite__remove__induct,axiom,
    ! [B: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,B)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ( A7 != bot_bo3990330152332043303nteger )
               => ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(A7),B)
                 => ( ! [X4: code_integer] :
                        ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A7)
                       => aa_set5643387332196575111eger_o(P,minus_2355218937544613996nteger(A7,insert_Code_integer(X4,bot_bo3990330152332043303nteger))) )
                   => aa_set5643387332196575111eger_o(P,A7) ) ) ) )
         => aa_set5643387332196575111eger_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_8147_finite__remove__induct,axiom,
    ! [B: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,B)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [A7: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,A7)
             => ( ( A7 != bot_bot_set_complex )
               => ( aa_set_complex_o(ord_le211207098394363844omplex(A7),B)
                 => ( ! [X4: complex] :
                        ( aa_set_complex_o(member_complex(X4),A7)
                       => aa_set_complex_o(P,minus_811609699411566653omplex(A7,insert_complex(X4,bot_bot_set_complex))) )
                   => aa_set_complex_o(P,A7) ) ) ) )
         => aa_set_complex_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_8148_finite__remove__induct,axiom,
    ! [B: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,B)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ( A7 != bot_bot_set_real )
               => ( aa_set_real_o(ord_less_eq_set_real(A7),B)
                 => ( ! [X4: real] :
                        ( aa_set_real_o(member_real(X4),A7)
                       => aa_set_real_o(P,minus_minus_set_real(A7,insert_real(X4,bot_bot_set_real))) )
                   => aa_set_real_o(P,A7) ) ) ) )
         => aa_set_real_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_8149_finite__remove__induct,axiom,
    ! [B: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,B)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ( A7 != bot_bot_set_o )
               => ( aa_set_o_o(ord_less_eq_set_o(A7),B)
                 => ( ! [X4: $o] :
                        ( aa_set_o_o(member_o((X4)),A7)
                       => aa_set_o_o(P,minus_minus_set_o(A7,insert_o((X4),bot_bot_set_o))) )
                   => aa_set_o_o(P,A7) ) ) ) )
         => aa_set_o_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_8150_finite__remove__induct,axiom,
    ! [B: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,B)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ( A7 != bot_bot_set_int )
               => ( aa_set_int_o(ord_less_eq_set_int(A7),B)
                 => ( ! [X4: int] :
                        ( aa_set_int_o(member_int(X4),A7)
                       => aa_set_int_o(P,minus_minus_set_int(A7,insert_int(X4,bot_bot_set_int))) )
                   => aa_set_int_o(P,A7) ) ) ) )
         => aa_set_int_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_8151_finite__remove__induct,axiom,
    ! [B: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,B)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ( A7 != bot_bot_set_nat )
               => ( aa_set_nat_o(ord_less_eq_set_nat(A7),B)
                 => ( ! [X4: nat] :
                        ( aa_set_nat_o(member_nat(X4),A7)
                       => aa_set_nat_o(P,minus_minus_set_nat(A7,insert_nat(X4,bot_bot_set_nat))) )
                   => aa_set_nat_o(P,A7) ) ) ) )
         => aa_set_nat_o(P,B) ) ) ) ).

% finite_remove_induct
tff(fact_8152_finite__induct__select,axiom,
    ! [S: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [T4: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(ord_le3480810397992357184T_VEBT(T4),S)
             => ( aa_set_VEBT_VEBT_o(P,T4)
               => ? [X4: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X4),minus_5127226145743854075T_VEBT(S,T4))
                    & aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X4,T4)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8153_finite__induct__select,axiom,
    ! [S: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [T4: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(ord_le1307284697595431911nteger(T4),S)
             => ( aa_set5643387332196575111eger_o(P,T4)
               => ? [X4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X4),minus_2355218937544613996nteger(S,T4))
                    & aa_set5643387332196575111eger_o(P,insert_Code_integer(X4,T4)) ) ) )
         => aa_set5643387332196575111eger_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8154_finite__induct__select,axiom,
    ! [S: set_complex,P: fun_set_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [T4: set_complex] :
              ( aa_set_complex_o(ord_less_set_complex(T4),S)
             => ( aa_set_complex_o(P,T4)
               => ? [X4: complex] :
                    ( aa_set_complex_o(member_complex(X4),minus_811609699411566653omplex(S,T4))
                    & aa_set_complex_o(P,insert_complex(X4,T4)) ) ) )
         => aa_set_complex_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8155_finite__induct__select,axiom,
    ! [S: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [T4: set_real] :
              ( aa_set_real_o(ord_less_set_real(T4),S)
             => ( aa_set_real_o(P,T4)
               => ? [X4: real] :
                    ( aa_set_real_o(member_real(X4),minus_minus_set_real(S,T4))
                    & aa_set_real_o(P,insert_real(X4,T4)) ) ) )
         => aa_set_real_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8156_finite__induct__select,axiom,
    ! [S: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [T4: set_o] :
              ( aa_set_o_o(ord_less_set_o(T4),S)
             => ( aa_set_o_o(P,T4)
               => ? [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),minus_minus_set_o(S,T4))
                    & aa_set_o_o(P,insert_o((X4),T4)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8157_finite__induct__select,axiom,
    ! [S: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [T4: set_int] :
              ( aa_set_int_o(ord_less_set_int(T4),S)
             => ( aa_set_int_o(P,T4)
               => ? [X4: int] :
                    ( aa_set_int_o(member_int(X4),minus_minus_set_int(S,T4))
                    & aa_set_int_o(P,insert_int(X4,T4)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8158_finite__induct__select,axiom,
    ! [S: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [T4: set_nat] :
              ( aa_set_nat_o(ord_less_set_nat(T4),S)
             => ( aa_set_nat_o(P,T4)
               => ? [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),minus_minus_set_nat(S,T4))
                    & aa_set_nat_o(P,insert_nat(X4,T4)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_induct_select
tff(fact_8159_minus__one__div__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% minus_one_div_numeral
tff(fact_8160_one__div__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% one_div_minus_numeral
tff(fact_8161_finite__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_gb(Nb),C2))) ) ).

% finite_nth_roots
tff(fact_8162_compl__le__compl__iff,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xc)),uminus5710092332889474511et_nat(Ya))
    <=> aa_set_nat_o(ord_less_eq_set_nat(Ya),Xc) ) ).

% compl_le_compl_iff
tff(fact_8163_Compl__iff,axiom,
    ! [C2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A))
    <=> ~ aa_set_nat_o(member_nat(C2),A) ) ).

% Compl_iff
tff(fact_8164_Compl__iff,axiom,
    ! [C2: real,A: set_real] :
      ( aa_set_real_o(member_real(C2),uminus612125837232591019t_real(A))
    <=> ~ aa_set_real_o(member_real(C2),A) ) ).

% Compl_iff
tff(fact_8165_Compl__iff,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),uminus8041839845116263051T_VEBT(A))
    <=> ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A) ) ).

% Compl_iff
tff(fact_8166_Compl__iff,axiom,
    ! [C2: int,A: set_int] :
      ( aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A))
    <=> ~ aa_set_int_o(member_int(C2),A) ) ).

% Compl_iff
tff(fact_8167_Compl__iff,axiom,
    ! [C2: complex,A: set_complex] :
      ( aa_set_complex_o(member_complex(C2),uminus8566677241136511917omplex(A))
    <=> ~ aa_set_complex_o(member_complex(C2),A) ) ).

% Compl_iff
tff(fact_8168_ComplI,axiom,
    ! [C2: nat,A: set_nat] :
      ( ~ aa_set_nat_o(member_nat(C2),A)
     => aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A)) ) ).

% ComplI
tff(fact_8169_ComplI,axiom,
    ! [C2: real,A: set_real] :
      ( ~ aa_set_real_o(member_real(C2),A)
     => aa_set_real_o(member_real(C2),uminus612125837232591019t_real(A)) ) ).

% ComplI
tff(fact_8170_ComplI,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A)
     => aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),uminus8041839845116263051T_VEBT(A)) ) ).

% ComplI
tff(fact_8171_ComplI,axiom,
    ! [C2: int,A: set_int] :
      ( ~ aa_set_int_o(member_int(C2),A)
     => aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A)) ) ).

% ComplI
tff(fact_8172_ComplI,axiom,
    ! [C2: complex,A: set_complex] :
      ( ~ aa_set_complex_o(member_complex(C2),A)
     => aa_set_complex_o(member_complex(C2),uminus8566677241136511917omplex(A)) ) ).

% ComplI
tff(fact_8173_minus__numeral__div__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Nb))) ).

% minus_numeral_div_numeral
tff(fact_8174_numeral__div__minus__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(M,Nb))) ).

% numeral_div_minus_numeral
tff(fact_8175_ComplD,axiom,
    ! [C2: nat,A: set_nat] :
      ( aa_set_nat_o(member_nat(C2),uminus5710092332889474511et_nat(A))
     => ~ aa_set_nat_o(member_nat(C2),A) ) ).

% ComplD
tff(fact_8176_ComplD,axiom,
    ! [C2: real,A: set_real] :
      ( aa_set_real_o(member_real(C2),uminus612125837232591019t_real(A))
     => ~ aa_set_real_o(member_real(C2),A) ) ).

% ComplD
tff(fact_8177_ComplD,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),uminus8041839845116263051T_VEBT(A))
     => ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(C2),A) ) ).

% ComplD
tff(fact_8178_ComplD,axiom,
    ! [C2: int,A: set_int] :
      ( aa_set_int_o(member_int(C2),uminus1532241313380277803et_int(A))
     => ~ aa_set_int_o(member_int(C2),A) ) ).

% ComplD
tff(fact_8179_ComplD,axiom,
    ! [C2: complex,A: set_complex] :
      ( aa_set_complex_o(member_complex(C2),uminus8566677241136511917omplex(A))
     => ~ aa_set_complex_o(member_complex(C2),A) ) ).

% ComplD
tff(fact_8180_Compl__eq,axiom,
    ! [A: set_real] : uminus612125837232591019t_real(A) = collect_real(aTP_Lamm_gc(A)) ).

% Compl_eq
tff(fact_8181_Compl__eq,axiom,
    ! [A: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(A) = collect_VEBT_VEBT(aTP_Lamm_gd(A)) ).

% Compl_eq
tff(fact_8182_Compl__eq,axiom,
    ! [A: set_nat] : uminus5710092332889474511et_nat(A) = collect_nat(aTP_Lamm_ge(A)) ).

% Compl_eq
tff(fact_8183_Compl__eq,axiom,
    ! [A: set_int] : uminus1532241313380277803et_int(A) = collect_int(aTP_Lamm_gf(A)) ).

% Compl_eq
tff(fact_8184_Compl__eq,axiom,
    ! [A: set_complex] : uminus8566677241136511917omplex(A) = collect_complex(aTP_Lamm_gg(A)) ).

% Compl_eq
tff(fact_8185_Compl__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int] : uminus6221592323253981072nt_int(A) = collec213857154873943460nt_int(aTP_Lamm_gh(A)) ).

% Compl_eq
tff(fact_8186_Collect__neg__eq,axiom,
    ! [P: fun_nat_o] : collect_nat(aTP_Lamm_gi(P)) = uminus5710092332889474511et_nat(collect_nat(P)) ).

% Collect_neg_eq
tff(fact_8187_Collect__neg__eq,axiom,
    ! [P: fun_int_o] : collect_int(aTP_Lamm_gj(P)) = uminus1532241313380277803et_int(collect_int(P)) ).

% Collect_neg_eq
tff(fact_8188_Collect__neg__eq,axiom,
    ! [P: fun_complex_o] : collect_complex(aTP_Lamm_gk(P)) = uminus8566677241136511917omplex(collect_complex(P)) ).

% Collect_neg_eq
tff(fact_8189_Collect__neg__eq,axiom,
    ! [P: fun_Pr3238374608701176162_int_o] : collec213857154873943460nt_int(aTP_Lamm_gl(P)) = uminus6221592323253981072nt_int(collec213857154873943460nt_int(P)) ).

% Collect_neg_eq
tff(fact_8190_uminus__set__def,axiom,
    ! [A: set_real] : uminus612125837232591019t_real(A) = collect_real(uminus2572480474538279996real_o(aTP_Lamm_a(A))) ).

% uminus_set_def
tff(fact_8191_uminus__set__def,axiom,
    ! [A: set_VEBT_VEBT] : uminus8041839845116263051T_VEBT(A) = collect_VEBT_VEBT(uminus3031588813561569520VEBT_o(aTP_Lamm_aa(A))) ).

% uminus_set_def
tff(fact_8192_uminus__set__def,axiom,
    ! [A: set_nat] : uminus5710092332889474511et_nat(A) = collect_nat(uminus6183504819137795864_nat_o(aTP_Lamm_ab(A))) ).

% uminus_set_def
tff(fact_8193_uminus__set__def,axiom,
    ! [A: set_int] : uminus1532241313380277803et_int(A) = collect_int(uminus4362462473705107900_int_o(aTP_Lamm_ac(A))) ).

% uminus_set_def
tff(fact_8194_uminus__set__def,axiom,
    ! [A: set_complex] : uminus8566677241136511917omplex(A) = collect_complex(uminus4035047602201070266plex_o(aTP_Lamm_ad(A))) ).

% uminus_set_def
tff(fact_8195_uminus__set__def,axiom,
    ! [A: set_Pr958786334691620121nt_int] : uminus6221592323253981072nt_int(A) = collec213857154873943460nt_int(uminus103585356917985579_int_o(aTP_Lamm_ae(A))) ).

% uminus_set_def
tff(fact_8196_compl__mono,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Ya)),uminus5710092332889474511et_nat(Xc)) ) ).

% compl_mono
tff(fact_8197_compl__le__swap1,axiom,
    ! [Ya: set_nat,Xc: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(Ya),uminus5710092332889474511et_nat(Xc))
     => aa_set_nat_o(ord_less_eq_set_nat(Xc),uminus5710092332889474511et_nat(Ya)) ) ).

% compl_le_swap1
tff(fact_8198_compl__le__swap2,axiom,
    ! [Ya: set_nat,Xc: set_nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Ya)),Xc)
     => aa_set_nat_o(ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xc)),Ya) ) ).

% compl_le_swap2
tff(fact_8199_complex__mod__minus__le__complex__mod,axiom,
    ! [Xc: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Xc))),aa_complex_real(real_V1022390504157884413omplex,Xc)) ).

% complex_mod_minus_le_complex_mod
tff(fact_8200_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(B2),A2))),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,A2)) ).

% complex_mod_triangle_ineq2
tff(fact_8201_diff__shunt__var,axiom,
    ! [Xc: set_real,Ya: set_real] :
      ( ( minus_minus_set_real(Xc,Ya) = bot_bot_set_real )
    <=> aa_set_real_o(ord_less_eq_set_real(Xc),Ya) ) ).

% diff_shunt_var
tff(fact_8202_diff__shunt__var,axiom,
    ! [Xc: set_o,Ya: set_o] :
      ( ( minus_minus_set_o(Xc,Ya) = bot_bot_set_o )
    <=> aa_set_o_o(ord_less_eq_set_o(Xc),Ya) ) ).

% diff_shunt_var
tff(fact_8203_diff__shunt__var,axiom,
    ! [Xc: set_int,Ya: set_int] :
      ( ( minus_minus_set_int(Xc,Ya) = bot_bot_set_int )
    <=> aa_set_int_o(ord_less_eq_set_int(Xc),Ya) ) ).

% diff_shunt_var
tff(fact_8204_diff__shunt__var,axiom,
    ! [Xc: set_nat,Ya: set_nat] :
      ( ( minus_minus_set_nat(Xc,Ya) = bot_bot_set_nat )
    <=> aa_set_nat_o(ord_less_eq_set_nat(Xc),Ya) ) ).

% diff_shunt_var
tff(fact_8205_set__encode__insert,axiom,
    ! [A: set_nat,Nb: nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( ~ aa_set_nat_o(member_nat(Nb),A)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_set_nat_nat(nat_set_encode,A)) ) ) ) ).

% set_encode_insert
tff(fact_8206_diff__preserves__multiset,axiom,
    ! [M3: fun_Pr5160692531401422620nt_nat,N5: fun_Pr5160692531401422620nt_nat] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_gm(M3)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun6302685600795359042_int_o(aTP_Lamm_gn(M3),N5))) ) ).

% diff_preserves_multiset
tff(fact_8207_diff__preserves__multiset,axiom,
    ! [M3: fun_nat_nat,N5: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_go(M3)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun8673969331427483654_nat_o(aTP_Lamm_gp(M3),N5))) ) ).

% diff_preserves_multiset
tff(fact_8208_diff__preserves__multiset,axiom,
    ! [M3: fun_int_nat,N5: fun_int_nat] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_gq(M3)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun5425778157628386638_int_o(aTP_Lamm_gr(M3),N5))) ) ).

% diff_preserves_multiset
tff(fact_8209_diff__preserves__multiset,axiom,
    ! [M3: fun_Code_integer_nat,N5: fun_Code_integer_nat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_gs(M3)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun4054834855839773292eger_o(aTP_Lamm_gt(M3),N5))) ) ).

% diff_preserves_multiset
tff(fact_8210_diff__preserves__multiset,axiom,
    ! [M3: fun_complex_nat,N5: fun_complex_nat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_gu(M3)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun8723867720292580170plex_o(aTP_Lamm_gv(M3),N5))) ) ).

% diff_preserves_multiset
tff(fact_8211_add__mset__in__multiset,axiom,
    ! [M3: fun_Pr5160692531401422620nt_nat,A2: product_prod_int_int] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_gm(M3)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_Pro3952998155787061599_int_o(aTP_Lamm_gw(M3),A2))) ) ).

% add_mset_in_multiset
tff(fact_8212_add__mset__in__multiset,axiom,
    ! [M3: fun_nat_nat,A2: nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_go(M3)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_nat_fun_nat_o(aTP_Lamm_gx(M3),A2))) ) ).

% add_mset_in_multiset
tff(fact_8213_add__mset__in__multiset,axiom,
    ! [M3: fun_int_nat,A2: int] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_gq(M3)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_int_fun_int_o(aTP_Lamm_gy(M3),A2))) ) ).

% add_mset_in_multiset
tff(fact_8214_add__mset__in__multiset,axiom,
    ! [M3: fun_Code_integer_nat,A2: code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_gs(M3)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_gz(M3),A2))) ) ).

% add_mset_in_multiset
tff(fact_8215_add__mset__in__multiset,axiom,
    ! [M3: fun_complex_nat,A2: complex] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_gu(M3)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_ha(M3),A2))) ) ).

% add_mset_in_multiset
tff(fact_8216_ln__series,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xc) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_hb,Xc)) ) ) ) ).

% ln_series
tff(fact_8217_dbl__dec__simps_I4_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_8218_dbl__dec__simps_I4_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_8219_dbl__dec__simps_I4_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_8220_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_8221_dbl__dec__simps_I4_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,one))) ).

% dbl_dec_simps(4)
tff(fact_8222_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_8223_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_8224_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_8225_set__encode__empty,axiom,
    aa_set_nat_nat(nat_set_encode,bot_bot_set_nat) = zero_zero_nat ).

% set_encode_empty
tff(fact_8226_dbl__dec__simps_I2_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,zero_zero_real) = aa_real_real(uminus_uminus_real,one_one_real) ).

% dbl_dec_simps(2)
tff(fact_8227_dbl__dec__simps_I2_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_dec_simps(2)
tff(fact_8228_dbl__dec__simps_I2_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,zero_zero_complex) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% dbl_dec_simps(2)
tff(fact_8229_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_8230_dbl__dec__simps_I2_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,zero_zero_rat) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_8231_powser__zero,axiom,
    ! [F2: fun_nat_complex] : suminf_complex(aTP_Lamm_hc(F2)) = aa_nat_complex(F2,zero_zero_nat) ).

% powser_zero
tff(fact_8232_powser__zero,axiom,
    ! [F2: fun_nat_real] : suminf_real(aTP_Lamm_hd(F2)) = aa_nat_real(F2,zero_zero_nat) ).

% powser_zero
tff(fact_8233_set__encode__inf,axiom,
    ! [A: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_nat(nat_set_encode,A) = zero_zero_nat ) ) ).

% set_encode_inf
tff(fact_8234_dbl__dec__def,axiom,
    ! [Xc: real] : aa_real_real(neg_nu6075765906172075777c_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Xc)),one_one_real) ).

% dbl_dec_def
tff(fact_8235_dbl__dec__def,axiom,
    ! [Xc: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,Xc) = aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(Xc),Xc)),one_one_rat) ).

% dbl_dec_def
tff(fact_8236_dbl__dec__def,axiom,
    ! [Xc: int] : aa_int_int(neg_nu3811975205180677377ec_int,Xc) = aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(Xc),Xc)),one_one_int) ).

% dbl_dec_def
tff(fact_8237_even__set__encode__iff,axiom,
    ! [A: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A))
      <=> ~ aa_set_nat_o(member_nat(zero_zero_nat),A) ) ) ).

% even_set_encode_iff
tff(fact_8238_filter__preserves__multiset,axiom,
    ! [M3: fun_Pr5160692531401422620nt_nat,P: fun_Pr3238374608701176162_int_o] :
      ( finite2998713641127702882nt_int(collec213857154873943460nt_int(aTP_Lamm_gm(M3)))
     => finite2998713641127702882nt_int(collec213857154873943460nt_int(aa_fun5349570537165229698_int_o(aTP_Lamm_he(M3),P))) ) ).

% filter_preserves_multiset
tff(fact_8239_filter__preserves__multiset,axiom,
    ! [M3: fun_nat_nat,P: fun_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(aTP_Lamm_go(M3)))
     => aa_set_nat_o(finite_finite_nat,collect_nat(aa_fun5937193402907943164_nat_o(aTP_Lamm_hf(M3),P))) ) ).

% filter_preserves_multiset
tff(fact_8240_filter__preserves__multiset,axiom,
    ! [M3: fun_int_nat,P: fun_int_o] :
      ( aa_set_int_o(finite_finite_int,collect_int(aTP_Lamm_gq(M3)))
     => aa_set_int_o(finite_finite_int,collect_int(aa_fun6243517787477317628_int_o(aTP_Lamm_hg(M3),P))) ) ).

% filter_preserves_multiset
tff(fact_8241_filter__preserves__multiset,axiom,
    ! [M3: fun_Code_integer_nat,P: fun_Code_integer_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aTP_Lamm_gs(M3)))
     => aa_set5643387332196575111eger_o(finite6017078050557962740nteger,collect_Code_integer(aa_fun8214818745163655228eger_o(aTP_Lamm_hh(M3),P))) ) ).

% filter_preserves_multiset
tff(fact_8242_filter__preserves__multiset,axiom,
    ! [M3: fun_complex_nat,P: fun_complex_o] :
      ( aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aTP_Lamm_gu(M3)))
     => aa_set_complex_o(finite3207457112153483333omplex,collect_complex(aa_fun2514317545224092796plex_o(aTP_Lamm_hi(M3),P))) ) ).

% filter_preserves_multiset
tff(fact_8243_suminf__geometric,axiom,
    ! [C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C2)),one_one_real)
     => ( suminf_real(aa_real_fun_nat_real(power_power_real,C2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C2)) ) ) ).

% suminf_geometric
tff(fact_8244_suminf__geometric,axiom,
    ! [C2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C2)),one_one_real)
     => ( suminf_complex(power_power_complex(C2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C2)) ) ) ).

% suminf_geometric
tff(fact_8245_suminf__zero,axiom,
    suminf_complex(aTP_Lamm_hj) = zero_zero_complex ).

% suminf_zero
tff(fact_8246_suminf__zero,axiom,
    suminf_real(aTP_Lamm_hk) = zero_zero_real ).

% suminf_zero
tff(fact_8247_suminf__zero,axiom,
    suminf_nat(aTP_Lamm_ct) = zero_zero_nat ).

% suminf_zero
tff(fact_8248_suminf__zero,axiom,
    suminf_int(aTP_Lamm_hl) = zero_zero_int ).

% suminf_zero
tff(fact_8249_pi__series,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = suminf_real(aTP_Lamm_hm) ).

% pi_series
tff(fact_8250_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc))),Xc))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_8251_finite__linorder__max__induct,axiom,
    ! [A: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B3: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ! [X4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A7)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X4),B3) )
               => ( aa_set5643387332196575111eger_o(P,A7)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(B3,A7)) ) ) )
         => aa_set5643387332196575111eger_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8252_finite__linorder__max__induct,axiom,
    ! [A: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B3: $o,A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ! [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),A7)
                   => aa_o_o(ord_less_o((X4)),(B3)) )
               => ( aa_set_o_o(P,A7)
                 => aa_set_o_o(P,insert_o((B3),A7)) ) ) )
         => aa_set_o_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8253_finite__linorder__max__induct,axiom,
    ! [A: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B3: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X4: real] :
                    ( aa_set_real_o(member_real(X4),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,X4),B3) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B3,A7)) ) ) )
         => aa_set_real_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8254_finite__linorder__max__induct,axiom,
    ! [A: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B3: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X4: rat] :
                    ( aa_set_rat_o(member_rat(X4),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),B3) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B3,A7)) ) ) )
         => aa_set_rat_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8255_finite__linorder__max__induct,axiom,
    ! [A: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B3: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X4: num] :
                    ( aa_set_num_o(member_num(X4),A7)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,X4),B3) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B3,A7)) ) ) )
         => aa_set_num_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8256_finite__linorder__max__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B3: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),B3) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B3,A7)) ) ) )
         => aa_set_nat_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8257_finite__linorder__max__induct,axiom,
    ! [A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B3: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X4: int] :
                    ( aa_set_int_o(member_int(X4),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,X4),B3) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B3,A7)) ) ) )
         => aa_set_int_o(P,A) ) ) ) ).

% finite_linorder_max_induct
tff(fact_8258_abs__idempotent,axiom,
    ! [A2: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A2)) = aa_real_real(abs_abs_real,A2) ).

% abs_idempotent
tff(fact_8259_abs__idempotent,axiom,
    ! [A2: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A2)) = aa_int_int(abs_abs_int,A2) ).

% abs_idempotent
tff(fact_8260_abs__idempotent,axiom,
    ! [A2: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A2)) = abs_abs_Code_integer(A2) ).

% abs_idempotent
tff(fact_8261_abs__idempotent,axiom,
    ! [A2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A2)) = aa_rat_rat(abs_abs_rat,A2) ).

% abs_idempotent
tff(fact_8262_abs__abs,axiom,
    ! [A2: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A2)) = aa_real_real(abs_abs_real,A2) ).

% abs_abs
tff(fact_8263_abs__abs,axiom,
    ! [A2: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A2)) = aa_int_int(abs_abs_int,A2) ).

% abs_abs
tff(fact_8264_abs__abs,axiom,
    ! [A2: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A2)) = abs_abs_Code_integer(A2) ).

% abs_abs
tff(fact_8265_abs__abs,axiom,
    ! [A2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A2)) = aa_rat_rat(abs_abs_rat,A2) ).

% abs_abs
tff(fact_8266_abs__0__eq,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A2) )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_8267_abs__0__eq,axiom,
    ! [A2: real] :
      ( ( zero_zero_real = aa_real_real(abs_abs_real,A2) )
    <=> ( A2 = zero_zero_real ) ) ).

% abs_0_eq
tff(fact_8268_abs__0__eq,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat = aa_rat_rat(abs_abs_rat,A2) )
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_8269_abs__0__eq,axiom,
    ! [A2: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A2) )
    <=> ( A2 = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_8270_abs__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer(A2) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_8271_abs__eq__0,axiom,
    ! [A2: real] :
      ( ( aa_real_real(abs_abs_real,A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% abs_eq_0
tff(fact_8272_abs__eq__0,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_8273_abs__eq__0,axiom,
    ! [A2: int] :
      ( ( aa_int_int(abs_abs_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_8274_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_8275_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_8276_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_8277_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_8278_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_8279_abs__0,axiom,
    aa_complex_complex(abs_abs_complex,zero_zero_complex) = zero_zero_complex ).

% abs_0
tff(fact_8280_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_8281_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_8282_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_8283_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(numera6620942414471956472nteger(Nb)) = numera6620942414471956472nteger(Nb) ).

% abs_numeral
tff(fact_8284_abs__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_numeral
tff(fact_8285_abs__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_numeral
tff(fact_8286_abs__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_numeral
tff(fact_8287_abs__add__abs,axiom,
    ! [A2: code_integer,B2: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B2))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B2)) ).

% abs_add_abs
tff(fact_8288_abs__add__abs,axiom,
    ! [A2: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2)) ).

% abs_add_abs
tff(fact_8289_abs__add__abs,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2))) = aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2)) ).

% abs_add_abs
tff(fact_8290_abs__add__abs,axiom,
    ! [A2: int,B2: int] : aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))) = aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2)) ).

% abs_add_abs
tff(fact_8291_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_8292_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_8293_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_8294_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_8295_abs__mult__self__eq,axiom,
    ! [A2: code_integer] : times_3573771949741848930nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(A2)) = times_3573771949741848930nteger(A2,A2) ).

% abs_mult_self_eq
tff(fact_8296_abs__mult__self__eq,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),A2) ).

% abs_mult_self_eq
tff(fact_8297_abs__mult__self__eq,axiom,
    ! [A2: rat] : aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,A2)) = aa_rat_rat(times_times_rat(A2),A2) ).

% abs_mult_self_eq
tff(fact_8298_abs__mult__self__eq,axiom,
    ! [A2: int] : aa_int_int(times_times_int(aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,A2)) = aa_int_int(times_times_int(A2),A2) ).

% abs_mult_self_eq
tff(fact_8299_abs__divide,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(abs_abs_complex,A2)),aa_complex_complex(abs_abs_complex,B2)) ).

% abs_divide
tff(fact_8300_abs__divide,axiom,
    ! [A2: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2)) ).

% abs_divide
tff(fact_8301_abs__divide,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A2),B2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2)) ).

% abs_divide
tff(fact_8302_abs__minus__cancel,axiom,
    ! [A2: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A2)) = aa_real_real(abs_abs_real,A2) ).

% abs_minus_cancel
tff(fact_8303_abs__minus__cancel,axiom,
    ! [A2: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A2)) = aa_int_int(abs_abs_int,A2) ).

% abs_minus_cancel
tff(fact_8304_abs__minus__cancel,axiom,
    ! [A2: code_integer] : abs_abs_Code_integer(uminus1351360451143612070nteger(A2)) = abs_abs_Code_integer(A2) ).

% abs_minus_cancel
tff(fact_8305_abs__minus__cancel,axiom,
    ! [A2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A2)) = aa_rat_rat(abs_abs_rat,A2) ).

% abs_minus_cancel
tff(fact_8306_abs__minus,axiom,
    ! [A2: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A2)) = aa_real_real(abs_abs_real,A2) ).

% abs_minus
tff(fact_8307_abs__minus,axiom,
    ! [A2: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A2)) = aa_int_int(abs_abs_int,A2) ).

% abs_minus
tff(fact_8308_abs__minus,axiom,
    ! [A2: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(uminus1482373934393186551omplex,A2)) = aa_complex_complex(abs_abs_complex,A2) ).

% abs_minus
tff(fact_8309_abs__minus,axiom,
    ! [A2: code_integer] : abs_abs_Code_integer(uminus1351360451143612070nteger(A2)) = abs_abs_Code_integer(A2) ).

% abs_minus
tff(fact_8310_abs__minus,axiom,
    ! [A2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A2)) = aa_rat_rat(abs_abs_rat,A2) ).

% abs_minus
tff(fact_8311_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),aa_real_real(abs_abs_real,K))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),K) ) ).

% dvd_abs_iff
tff(fact_8312_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),aa_int_int(abs_abs_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% dvd_abs_iff
tff(fact_8313_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),abs_abs_Code_integer(K))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% dvd_abs_iff
tff(fact_8314_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),aa_rat_rat(abs_abs_rat,K))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% dvd_abs_iff
tff(fact_8315_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(abs_abs_real,M)),K)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,M),K) ) ).

% abs_dvd_iff
tff(fact_8316_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(abs_abs_int,M)),K)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,M),K) ) ).

% abs_dvd_iff
tff(fact_8317_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,abs_abs_Code_integer(M)),K)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,M),K) ) ).

% abs_dvd_iff
tff(fact_8318_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(abs_abs_rat,M)),K)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,M),K) ) ).

% abs_dvd_iff
tff(fact_8319_abs__of__nat,axiom,
    ! [Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)) = aa_nat_rat(semiri681578069525770553at_rat,Nb) ).

% abs_of_nat
tff(fact_8320_abs__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(semiri5074537144036343181t_real,Nb) ).

% abs_of_nat
tff(fact_8321_abs__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_nat_int(semiri1314217659103216013at_int,Nb) ).

% abs_of_nat
tff(fact_8322_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_Code_integer(aa_nat_Code_integer(semiri4939895301339042750nteger,Nb)) = aa_nat_Code_integer(semiri4939895301339042750nteger,Nb) ).

% abs_of_nat
tff(fact_8323_abs__of__nonneg,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
     => ( abs_abs_Code_integer(A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_8324_abs__of__nonneg,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_real(abs_abs_real,A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_8325_abs__of__nonneg,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
     => ( aa_rat_rat(abs_abs_rat,A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_8326_abs__of__nonneg,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_int(abs_abs_int,A2) = A2 ) ) ).

% abs_of_nonneg
tff(fact_8327_abs__le__self__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),A2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2) ) ).

% abs_le_self_iff
tff(fact_8328_abs__le__self__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2) ) ).

% abs_le_self_iff
tff(fact_8329_abs__le__self__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2) ) ).

% abs_le_self_iff
tff(fact_8330_abs__le__self__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2) ) ).

% abs_le_self_iff
tff(fact_8331_abs__le__zero__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),zero_z3403309356797280102nteger)
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_8332_abs__le__zero__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),zero_zero_real)
    <=> ( A2 = zero_zero_real ) ) ).

% abs_le_zero_iff
tff(fact_8333_abs__le__zero__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),zero_zero_rat)
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_8334_abs__le__zero__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),zero_zero_int)
    <=> ( A2 = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_8335_zero__less__abs__iff,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A2))
    <=> ( A2 != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_8336_zero__less__abs__iff,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,A2))
    <=> ( A2 != zero_zero_real ) ) ).

% zero_less_abs_iff
tff(fact_8337_zero__less__abs__iff,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A2))
    <=> ( A2 != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_8338_zero__less__abs__iff,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A2))
    <=> ( A2 != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_8339_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,Nb) ).

% abs_neg_numeral
tff(fact_8340_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Nb) ).

% abs_neg_numeral
tff(fact_8341_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(uminus1351360451143612070nteger(numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(Nb) ).

% abs_neg_numeral
tff(fact_8342_abs__neg__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,Nb) ).

% abs_neg_numeral
tff(fact_8343_abs__neg__one,axiom,
    aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,one_one_real)) = one_one_real ).

% abs_neg_one
tff(fact_8344_abs__neg__one,axiom,
    aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,one_one_int)) = one_one_int ).

% abs_neg_one
tff(fact_8345_abs__neg__one,axiom,
    abs_abs_Code_integer(uminus1351360451143612070nteger(one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_8346_abs__neg__one,axiom,
    aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_8347_abs__power__minus,axiom,
    ! [A2: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,A2)),Nb)) = aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) ).

% abs_power_minus
tff(fact_8348_abs__power__minus,axiom,
    ! [A2: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A2)),Nb)) = aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A2),Nb)) ).

% abs_power_minus
tff(fact_8349_abs__power__minus,axiom,
    ! [A2: code_integer,Nb: nat] : abs_abs_Code_integer(power_8256067586552552935nteger(uminus1351360451143612070nteger(A2),Nb)) = abs_abs_Code_integer(power_8256067586552552935nteger(A2,Nb)) ).

% abs_power_minus
tff(fact_8350_abs__power__minus,axiom,
    ! [A2: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A2)),Nb)) = aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A2),Nb)) ).

% abs_power_minus
tff(fact_8351_zero__le__divide__abs__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(abs_abs_real,B2)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
        | ( B2 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_8352_zero__le__divide__abs__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(abs_abs_rat,B2)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
        | ( B2 = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_8353_divide__le__0__abs__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(abs_abs_real,B2))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
        | ( B2 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_8354_divide__le__0__abs__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A2),aa_rat_rat(abs_abs_rat,B2))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_8355_abs__of__nonpos,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A2) = aa_real_real(uminus_uminus_real,A2) ) ) ).

% abs_of_nonpos
tff(fact_8356_abs__of__nonpos,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A2) = uminus1351360451143612070nteger(A2) ) ) ).

% abs_of_nonpos
tff(fact_8357_abs__of__nonpos,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A2) = aa_rat_rat(uminus_uminus_rat,A2) ) ) ).

% abs_of_nonpos
tff(fact_8358_abs__of__nonpos,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A2) = aa_int_int(uminus_uminus_int,A2) ) ) ).

% abs_of_nonpos
tff(fact_8359_artanh__minus__real,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,Xc)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,Xc)) ) ) ).

% artanh_minus_real
tff(fact_8360_zero__less__power__abs__iff,axiom,
    ! [A2: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(abs_abs_Code_integer(A2),Nb))
    <=> ( ( A2 != zero_z3403309356797280102nteger )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8361_zero__less__power__abs__iff,axiom,
    ! [A2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),Nb))
    <=> ( ( A2 != zero_zero_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8362_zero__less__power__abs__iff,axiom,
    ! [A2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A2)),Nb))
    <=> ( ( A2 != zero_zero_rat )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8363_zero__less__power__abs__iff,axiom,
    ! [A2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),Nb))
    <=> ( ( A2 != zero_zero_int )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_8364_power2__abs,axiom,
    ! [A2: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_8365_power2__abs,axiom,
    ! [A2: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_8366_power2__abs,axiom,
    ! [A2: int] : aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_8367_power2__abs,axiom,
    ! [A2: code_integer] : power_8256067586552552935nteger(abs_abs_Code_integer(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_8368_abs__power2,axiom,
    ! [A2: rat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_8369_abs__power2,axiom,
    ! [A2: real] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_8370_abs__power2,axiom,
    ! [A2: int] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_8371_abs__power2,axiom,
    ! [A2: code_integer] : abs_abs_Code_integer(power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_8372_power__even__abs__numeral,axiom,
    ! [W: num,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A2)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_rat(power_power_rat(A2),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_8373_power__even__abs__numeral,axiom,
    ! [W: num,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_8374_power__even__abs__numeral,axiom,
    ! [W: num,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),aa_num_nat(numeral_numeral_nat,W)) = aa_nat_int(power_power_int(A2),aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_8375_power__even__abs__numeral,axiom,
    ! [W: num,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,W))
     => ( power_8256067586552552935nteger(abs_abs_Code_integer(A2),aa_num_nat(numeral_numeral_nat,W)) = power_8256067586552552935nteger(A2,aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_8376_abs__eq__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( ( aa_real_real(abs_abs_real,Xc) = aa_real_real(abs_abs_real,Ya) )
    <=> ( ( Xc = Ya )
        | ( Xc = aa_real_real(uminus_uminus_real,Ya) ) ) ) ).

% abs_eq_iff
tff(fact_8377_abs__eq__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( ( aa_int_int(abs_abs_int,Xc) = aa_int_int(abs_abs_int,Ya) )
    <=> ( ( Xc = Ya )
        | ( Xc = aa_int_int(uminus_uminus_int,Ya) ) ) ) ).

% abs_eq_iff
tff(fact_8378_abs__eq__iff,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( ( abs_abs_Code_integer(Xc) = abs_abs_Code_integer(Ya) )
    <=> ( ( Xc = Ya )
        | ( Xc = uminus1351360451143612070nteger(Ya) ) ) ) ).

% abs_eq_iff
tff(fact_8379_abs__eq__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( ( aa_rat_rat(abs_abs_rat,Xc) = aa_rat_rat(abs_abs_rat,Ya) )
    <=> ( ( Xc = Ya )
        | ( Xc = aa_rat_rat(uminus_uminus_rat,Ya) ) ) ) ).

% abs_eq_iff
tff(fact_8380_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
tff(fact_8381_abs__ge__self,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),aa_real_real(abs_abs_real,A2)) ).

% abs_ge_self
tff(fact_8382_abs__ge__self,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),abs_abs_Code_integer(A2)) ).

% abs_ge_self
tff(fact_8383_abs__ge__self,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),aa_rat_rat(abs_abs_rat,A2)) ).

% abs_ge_self
tff(fact_8384_abs__ge__self,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),aa_int_int(abs_abs_int,A2)) ).

% abs_ge_self
tff(fact_8385_abs__le__D1,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ).

% abs_le_D1
tff(fact_8386_abs__le__D1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2) ) ).

% abs_le_D1
tff(fact_8387_abs__le__D1,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2) ) ).

% abs_le_D1
tff(fact_8388_abs__le__D1,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2) ) ).

% abs_le_D1
tff(fact_8389_abs__mult,axiom,
    ! [A2: code_integer,B2: code_integer] : abs_abs_Code_integer(times_3573771949741848930nteger(A2,B2)) = times_3573771949741848930nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B2)) ).

% abs_mult
tff(fact_8390_abs__mult,axiom,
    ! [A2: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2)) ).

% abs_mult
tff(fact_8391_abs__mult,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A2),B2)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2)) ).

% abs_mult
tff(fact_8392_abs__mult,axiom,
    ! [A2: int,B2: int] : aa_int_int(abs_abs_int,aa_int_int(times_times_int(A2),B2)) = aa_int_int(times_times_int(aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2)) ).

% abs_mult
tff(fact_8393_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_8394_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_8395_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_8396_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_8397_abs__minus__commute,axiom,
    ! [A2: code_integer,B2: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A2,B2)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B2,A2)) ).

% abs_minus_commute
tff(fact_8398_abs__minus__commute,axiom,
    ! [A2: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)) ).

% abs_minus_commute
tff(fact_8399_abs__minus__commute,axiom,
    ! [A2: rat,B2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A2),B2)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B2),A2)) ).

% abs_minus_commute
tff(fact_8400_abs__minus__commute,axiom,
    ! [A2: int,B2: int] : aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A2),B2)) = aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B2),A2)) ).

% abs_minus_commute
tff(fact_8401_abs__eq__0__iff,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer(A2) = zero_z3403309356797280102nteger )
    <=> ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_8402_abs__eq__0__iff,axiom,
    ! [A2: complex] :
      ( ( aa_complex_complex(abs_abs_complex,A2) = zero_zero_complex )
    <=> ( A2 = zero_zero_complex ) ) ).

% abs_eq_0_iff
tff(fact_8403_abs__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( aa_real_real(abs_abs_real,A2) = zero_zero_real )
    <=> ( A2 = zero_zero_real ) ) ).

% abs_eq_0_iff
tff(fact_8404_abs__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A2) = zero_zero_rat )
    <=> ( A2 = zero_zero_rat ) ) ).

% abs_eq_0_iff
tff(fact_8405_abs__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( aa_int_int(abs_abs_int,A2) = zero_zero_int )
    <=> ( A2 = zero_zero_int ) ) ).

% abs_eq_0_iff
tff(fact_8406_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),K) ) ).

% dvd_if_abs_eq
tff(fact_8407_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K) ) ).

% dvd_if_abs_eq
tff(fact_8408_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),K) ) ).

% dvd_if_abs_eq
tff(fact_8409_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( aa_rat_rat(abs_abs_rat,L) = aa_rat_rat(abs_abs_rat,K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),K) ) ).

% dvd_if_abs_eq
tff(fact_8410_power__abs,axiom,
    ! [A2: rat,Nb: nat] : aa_rat_rat(abs_abs_rat,aa_nat_rat(power_power_rat(A2),Nb)) = aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A2)),Nb) ).

% power_abs
tff(fact_8411_power__abs,axiom,
    ! [A2: real,Nb: nat] : aa_real_real(abs_abs_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),Nb) ).

% power_abs
tff(fact_8412_power__abs,axiom,
    ! [A2: int,Nb: nat] : aa_int_int(abs_abs_int,aa_nat_int(power_power_int(A2),Nb)) = aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),Nb) ).

% power_abs
tff(fact_8413_power__abs,axiom,
    ! [A2: code_integer,Nb: nat] : abs_abs_Code_integer(power_8256067586552552935nteger(A2,Nb)) = power_8256067586552552935nteger(abs_abs_Code_integer(A2),Nb) ).

% power_abs
tff(fact_8414_abs__ge__zero,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),abs_abs_Code_integer(A2)) ).

% abs_ge_zero
tff(fact_8415_abs__ge__zero,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(abs_abs_real,A2)) ).

% abs_ge_zero
tff(fact_8416_abs__ge__zero,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(abs_abs_rat,A2)) ).

% abs_ge_zero
tff(fact_8417_abs__ge__zero,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A2)) ).

% abs_ge_zero
tff(fact_8418_abs__not__less__zero,axiom,
    ! [A2: code_integer] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A2)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_8419_abs__not__less__zero,axiom,
    ! [A2: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A2)),zero_zero_real) ).

% abs_not_less_zero
tff(fact_8420_abs__not__less__zero,axiom,
    ! [A2: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A2)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_8421_abs__not__less__zero,axiom,
    ! [A2: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A2)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_8422_abs__of__pos,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A2)
     => ( abs_abs_Code_integer(A2) = A2 ) ) ).

% abs_of_pos
tff(fact_8423_abs__of__pos,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_real(abs_abs_real,A2) = A2 ) ) ).

% abs_of_pos
tff(fact_8424_abs__of__pos,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2)
     => ( aa_rat_rat(abs_abs_rat,A2) = A2 ) ) ).

% abs_of_pos
tff(fact_8425_abs__of__pos,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A2)
     => ( aa_int_int(abs_abs_int,A2) = A2 ) ) ).

% abs_of_pos
tff(fact_8426_abs__triangle__ineq,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A2,B2))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B2))) ).

% abs_triangle_ineq
tff(fact_8427_abs__triangle__ineq,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))) ).

% abs_triangle_ineq
tff(fact_8428_abs__triangle__ineq,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(plus_plus_rat(A2),B2))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2))) ).

% abs_triangle_ineq
tff(fact_8429_abs__triangle__ineq,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(A2),B2))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))) ).

% abs_triangle_ineq
tff(fact_8430_abs__mult__less,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A2)),C2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(B2)),D2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,times_3573771949741848930nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B2))),times_3573771949741848930nteger(C2,D2)) ) ) ).

% abs_mult_less
tff(fact_8431_abs__mult__less,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A2)),C2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,B2)),D2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C2),D2)) ) ) ).

% abs_mult_less
tff(fact_8432_abs__mult__less,axiom,
    ! [A2: rat,C2: rat,B2: rat,D2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A2)),C2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,B2)),D2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2))),aa_rat_rat(times_times_rat(C2),D2)) ) ) ).

% abs_mult_less
tff(fact_8433_abs__mult__less,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A2)),C2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B2)),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))),aa_int_int(times_times_int(C2),D2)) ) ) ).

% abs_mult_less
tff(fact_8434_abs__triangle__ineq2,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B2))),abs_abs_Code_integer(minus_8373710615458151222nteger(A2,B2))) ).

% abs_triangle_ineq2
tff(fact_8435_abs__triangle__ineq2,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))) ).

% abs_triangle_ineq2
tff(fact_8436_abs__triangle__ineq2,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A2),B2))) ).

% abs_triangle_ineq2
tff(fact_8437_abs__triangle__ineq2,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A2),B2))) ).

% abs_triangle_ineq2
tff(fact_8438_abs__triangle__ineq3,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B2)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A2,B2))) ).

% abs_triangle_ineq3
tff(fact_8439_abs__triangle__ineq3,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))) ).

% abs_triangle_ineq3
tff(fact_8440_abs__triangle__ineq3,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A2),B2))) ).

% abs_triangle_ineq3
tff(fact_8441_abs__triangle__ineq3,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2)))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A2),B2))) ).

% abs_triangle_ineq3
tff(fact_8442_abs__triangle__ineq2__sym,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B2))),abs_abs_Code_integer(minus_8373710615458151222nteger(B2,A2))) ).

% abs_triangle_ineq2_sym
tff(fact_8443_abs__triangle__ineq2__sym,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2))) ).

% abs_triangle_ineq2_sym
tff(fact_8444_abs__triangle__ineq2__sym,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B2),A2))) ).

% abs_triangle_ineq2_sym
tff(fact_8445_abs__triangle__ineq2__sym,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B2),A2))) ).

% abs_triangle_ineq2_sym
tff(fact_8446_nonzero__abs__divide,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2)) ) ) ).

% nonzero_abs_divide
tff(fact_8447_nonzero__abs__divide,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A2),B2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2)) ) ) ).

% nonzero_abs_divide
tff(fact_8448_abs__ge__minus__self,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),aa_real_real(abs_abs_real,A2)) ).

% abs_ge_minus_self
tff(fact_8449_abs__ge__minus__self,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A2)),abs_abs_Code_integer(A2)) ).

% abs_ge_minus_self
tff(fact_8450_abs__ge__minus__self,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),aa_rat_rat(abs_abs_rat,A2)) ).

% abs_ge_minus_self
tff(fact_8451_abs__ge__minus__self,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),aa_int_int(abs_abs_int,A2)) ).

% abs_ge_minus_self
tff(fact_8452_abs__le__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B2) ) ) ).

% abs_le_iff
tff(fact_8453_abs__le__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),B2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A2)),B2) ) ) ).

% abs_le_iff
tff(fact_8454_abs__le__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),B2) ) ) ).

% abs_le_iff
tff(fact_8455_abs__le__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B2) ) ) ).

% abs_le_iff
tff(fact_8456_abs__le__D2,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B2) ) ).

% abs_le_D2
tff(fact_8457_abs__le__D2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A2)),B2) ) ).

% abs_le_D2
tff(fact_8458_abs__le__D2,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),B2) ) ).

% abs_le_D2
tff(fact_8459_abs__le__D2,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B2) ) ).

% abs_le_D2
tff(fact_8460_abs__leI,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A2)),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),B2) ) ) ).

% abs_leI
tff(fact_8461_abs__leI,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(A2)),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),B2) ) ) ).

% abs_leI
tff(fact_8462_abs__leI,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A2)),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),B2) ) ) ).

% abs_leI
tff(fact_8463_abs__leI,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A2)),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),B2) ) ) ).

% abs_leI
tff(fact_8464_abs__less__iff,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A2)),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A2)),B2) ) ) ).

% abs_less_iff
tff(fact_8465_abs__less__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A2)),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A2)),B2) ) ) ).

% abs_less_iff
tff(fact_8466_abs__less__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A2)),B2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),B2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,uminus1351360451143612070nteger(A2)),B2) ) ) ).

% abs_less_iff
tff(fact_8467_abs__less__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A2)),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A2)),B2) ) ) ).

% abs_less_iff
tff(fact_8468_pi__not__less__zero,axiom,
    ~ aa_real_o(aa_real_fun_real_o(ord_less_real,pi),zero_zero_real) ).

% pi_not_less_zero
tff(fact_8469_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_8470_pi__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),pi) ).

% pi_ge_zero
tff(fact_8471_dense__eq0__I,axiom,
    ! [Xc: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),E2) )
     => ( Xc = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_8472_dense__eq0__I,axiom,
    ! [Xc: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xc)),E2) )
     => ( Xc = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_8473_abs__eq__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A2),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(times_3573771949741848930nteger(A2,B2)) = times_3573771949741848930nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B2)) ) ) ).

% abs_eq_mult
tff(fact_8474_abs__eq__mult,axiom,
    ! [A2: real,B2: real] :
      ( ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2)) ) ) ).

% abs_eq_mult
tff(fact_8475_abs__eq__mult,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A2),zero_zero_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(A2),B2)) = aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2)) ) ) ).

% abs_eq_mult
tff(fact_8476_abs__eq__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(times_times_int(A2),B2)) = aa_int_int(times_times_int(aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2)) ) ) ).

% abs_eq_mult
tff(fact_8477_abs__mult__pos,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xc)
     => ( times_3573771949741848930nteger(abs_abs_Code_integer(Ya),Xc) = abs_abs_Code_integer(times_3573771949741848930nteger(Ya,Xc)) ) ) ).

% abs_mult_pos
tff(fact_8478_abs__mult__pos,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Ya)),Xc) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Xc)) ) ) ).

% abs_mult_pos
tff(fact_8479_abs__mult__pos,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xc)
     => ( aa_rat_rat(times_times_rat(aa_rat_rat(abs_abs_rat,Ya)),Xc) = aa_rat_rat(abs_abs_rat,aa_rat_rat(times_times_rat(Ya),Xc)) ) ) ).

% abs_mult_pos
tff(fact_8480_abs__mult__pos,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_int(times_times_int(aa_int_int(abs_abs_int,Ya)),Xc) = aa_int_int(abs_abs_int,aa_int_int(times_times_int(Ya),Xc)) ) ) ).

% abs_mult_pos
tff(fact_8481_abs__div__pos,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,Xc)),Ya) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) ) ) ).

% abs_div_pos
tff(fact_8482_abs__div__pos,axiom,
    ! [Ya: rat,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Ya)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Xc)),Ya) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Xc),Ya)) ) ) ).

% abs_div_pos
tff(fact_8483_zero__le__power__abs,axiom,
    ! [A2: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),Nb)) ).

% zero_le_power_abs
tff(fact_8484_zero__le__power__abs,axiom,
    ! [A2: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(abs_abs_Code_integer(A2),Nb)) ).

% zero_le_power_abs
tff(fact_8485_zero__le__power__abs,axiom,
    ! [A2: rat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A2)),Nb)) ).

% zero_le_power_abs
tff(fact_8486_zero__le__power__abs,axiom,
    ! [A2: int,Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),Nb)) ).

% zero_le_power_abs
tff(fact_8487_abs__minus__le__zero,axiom,
    ! [A2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(abs_abs_real,A2))),zero_zero_real) ).

% abs_minus_le_zero
tff(fact_8488_abs__minus__le__zero,axiom,
    ! [A2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,uminus1351360451143612070nteger(abs_abs_Code_integer(A2))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_8489_abs__minus__le__zero,axiom,
    ! [A2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(abs_abs_rat,A2))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_8490_abs__minus__le__zero,axiom,
    ! [A2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,A2))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_8491_eq__abs__iff_H,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = aa_real_real(abs_abs_real,B2) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
        & ( ( B2 = A2 )
          | ( B2 = aa_real_real(uminus_uminus_real,A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_8492_eq__abs__iff_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 = abs_abs_Code_integer(B2) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A2)
        & ( ( B2 = A2 )
          | ( B2 = uminus1351360451143612070nteger(A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_8493_eq__abs__iff_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = aa_rat_rat(abs_abs_rat,B2) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A2)
        & ( ( B2 = A2 )
          | ( B2 = aa_rat_rat(uminus_uminus_rat,A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_8494_eq__abs__iff_H,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = aa_int_int(abs_abs_int,B2) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
        & ( ( B2 = A2 )
          | ( B2 = aa_int_int(uminus_uminus_int,A2) ) ) ) ) ).

% eq_abs_iff'
tff(fact_8495_abs__eq__iff_H,axiom,
    ! [A2: real,B2: real] :
      ( ( aa_real_real(abs_abs_real,A2) = B2 )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
        & ( ( A2 = B2 )
          | ( A2 = aa_real_real(uminus_uminus_real,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_8496_abs__eq__iff_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer(A2) = B2 )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
        & ( ( A2 = B2 )
          | ( A2 = uminus1351360451143612070nteger(B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_8497_abs__eq__iff_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A2) = B2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
        & ( ( A2 = B2 )
          | ( A2 = aa_rat_rat(uminus_uminus_rat,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_8498_abs__eq__iff_H,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(abs_abs_int,A2) = B2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
        & ( ( A2 = B2 )
          | ( A2 = aa_int_int(uminus_uminus_int,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_8499_abs__if__raw,axiom,
    ! [X4: real] :
      aa_real_real(abs_abs_real,X4) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X4),zero_zero_real),aa_real_real(uminus_uminus_real,X4),X4) ).

% abs_if_raw
tff(fact_8500_abs__if__raw,axiom,
    ! [X4: int] :
      aa_int_int(abs_abs_int,X4) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X4),zero_zero_int),aa_int_int(uminus_uminus_int,X4),X4) ).

% abs_if_raw
tff(fact_8501_abs__if__raw,axiom,
    ! [X4: code_integer] :
      abs_abs_Code_integer(X4) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X4),zero_z3403309356797280102nteger),uminus1351360451143612070nteger(X4),X4) ).

% abs_if_raw
tff(fact_8502_abs__if__raw,axiom,
    ! [X4: rat] :
      aa_rat_rat(abs_abs_rat,X4) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X4),X4) ).

% abs_if_raw
tff(fact_8503_abs__of__neg,axiom,
    ! [A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_real(abs_abs_real,A2) = aa_real_real(uminus_uminus_real,A2) ) ) ).

% abs_of_neg
tff(fact_8504_abs__of__neg,axiom,
    ! [A2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A2) = aa_int_int(uminus_uminus_int,A2) ) ) ).

% abs_of_neg
tff(fact_8505_abs__of__neg,axiom,
    ! [A2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A2) = uminus1351360451143612070nteger(A2) ) ) ).

% abs_of_neg
tff(fact_8506_abs__of__neg,axiom,
    ! [A2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat)
     => ( aa_rat_rat(abs_abs_rat,A2) = aa_rat_rat(uminus_uminus_rat,A2) ) ) ).

% abs_of_neg
tff(fact_8507_abs__if,axiom,
    ! [A2: real] :
      aa_real_real(abs_abs_real,A2) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real),aa_real_real(uminus_uminus_real,A2),A2) ).

% abs_if
tff(fact_8508_abs__if,axiom,
    ! [A2: int] :
      aa_int_int(abs_abs_int,A2) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,A2),zero_zero_int),aa_int_int(uminus_uminus_int,A2),A2) ).

% abs_if
tff(fact_8509_abs__if,axiom,
    ! [A2: code_integer] :
      abs_abs_Code_integer(A2) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A2),zero_z3403309356797280102nteger),uminus1351360451143612070nteger(A2),A2) ).

% abs_if
tff(fact_8510_abs__if,axiom,
    ! [A2: rat] :
      aa_rat_rat(abs_abs_rat,A2) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A2),A2) ).

% abs_if
tff(fact_8511_abs__diff__le__iff,axiom,
    ! [Xc: code_integer,A2: code_integer,R3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xc,A2))),R3)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A2,R3)),Xc)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xc),plus_p5714425477246183910nteger(A2,R3)) ) ) ).

% abs_diff_le_iff
tff(fact_8512_abs__diff__le__iff,axiom,
    ! [Xc: real,A2: real,R3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),A2))),R3)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),R3)),Xc)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),R3)) ) ) ).

% abs_diff_le_iff
tff(fact_8513_abs__diff__le__iff,axiom,
    ! [Xc: rat,A2: rat,R3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xc),A2))),R3)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A2),R3)),Xc)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),aa_rat_rat(plus_plus_rat(A2),R3)) ) ) ).

% abs_diff_le_iff
tff(fact_8514_abs__diff__le__iff,axiom,
    ! [Xc: int,A2: int,R3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xc),A2))),R3)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(minus_minus_int(A2),R3)),Xc)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xc),aa_int_int(plus_plus_int(A2),R3)) ) ) ).

% abs_diff_le_iff
tff(fact_8515_abs__diff__triangle__ineq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A2,B2),plus_p5714425477246183910nteger(C2,D2)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A2,C2)),abs_abs_Code_integer(minus_8373710615458151222nteger(B2,D2)))) ).

% abs_diff_triangle_ineq
tff(fact_8516_abs__diff__triangle__ineq,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C2),D2)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_8517_abs__diff__triangle__ineq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(plus_plus_rat(A2),B2)),aa_rat_rat(plus_plus_rat(C2),D2)))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A2),C2))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B2),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_8518_abs__diff__triangle__ineq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(plus_plus_int(A2),B2)),aa_int_int(plus_plus_int(C2),D2)))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A2),C2))),aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(B2),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_8519_abs__triangle__ineq4,axiom,
    ! [A2: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A2,B2))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A2),abs_abs_Code_integer(B2))) ).

% abs_triangle_ineq4
tff(fact_8520_abs__triangle__ineq4,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))) ).

% abs_triangle_ineq4
tff(fact_8521_abs__triangle__ineq4,axiom,
    ! [A2: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A2),B2))),aa_rat_rat(plus_plus_rat(aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2))) ).

% abs_triangle_ineq4
tff(fact_8522_abs__triangle__ineq4,axiom,
    ! [A2: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(A2),B2))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))) ).

% abs_triangle_ineq4
tff(fact_8523_abs__diff__less__iff,axiom,
    ! [Xc: code_integer,A2: code_integer,R3: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xc,A2))),R3)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A2,R3)),Xc)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xc),plus_p5714425477246183910nteger(A2,R3)) ) ) ).

% abs_diff_less_iff
tff(fact_8524_abs__diff__less__iff,axiom,
    ! [Xc: real,A2: real,R3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),A2))),R3)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),R3)),Xc)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),R3)) ) ) ).

% abs_diff_less_iff
tff(fact_8525_abs__diff__less__iff,axiom,
    ! [Xc: rat,A2: rat,R3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xc),A2))),R3)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A2),R3)),Xc)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),aa_rat_rat(plus_plus_rat(A2),R3)) ) ) ).

% abs_diff_less_iff
tff(fact_8526_abs__diff__less__iff,axiom,
    ! [Xc: int,A2: int,R3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xc),A2))),R3)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(A2),R3)),Xc)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),aa_int_int(plus_plus_int(A2),R3)) ) ) ).

% abs_diff_less_iff
tff(fact_8527_abs__real__def,axiom,
    ! [A2: real] :
      aa_real_real(abs_abs_real,A2) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real),aa_real_real(uminus_uminus_real,A2),A2) ).

% abs_real_def
tff(fact_8528_lemma__interval__lt,axiom,
    ! [A2: real,Xc: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),B2)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [Y3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Y3))),D5)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_8529_sin__bound__lemma,axiom,
    ! [Xc: real,Ya: real,U: real,V: real] :
      ( ( Xc = Ya )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),U)),Ya))),V) ) ) ).

% sin_bound_lemma
tff(fact_8530_abs__add__one__gt__zero,axiom,
    ! [Xc: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xc))) ).

% abs_add_one_gt_zero
tff(fact_8531_abs__add__one__gt__zero,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,Xc))) ).

% abs_add_one_gt_zero
tff(fact_8532_abs__add__one__gt__zero,axiom,
    ! [Xc: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(plus_plus_rat(one_one_rat),aa_rat_rat(abs_abs_rat,Xc))) ).

% abs_add_one_gt_zero
tff(fact_8533_abs__add__one__gt__zero,axiom,
    ! [Xc: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(plus_plus_int(one_one_int),aa_int_int(abs_abs_int,Xc))) ).

% abs_add_one_gt_zero
tff(fact_8534_of__int__leD,axiom,
    ! [Nb: int,Xc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(ring_18347121197199848620nteger(Nb))),Xc)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xc) ) ) ).

% of_int_leD
tff(fact_8535_of__int__leD,axiom,
    ! [Nb: int,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xc)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc) ) ) ).

% of_int_leD
tff(fact_8536_of__int__leD,axiom,
    ! [Nb: int,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xc)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xc) ) ) ).

% of_int_leD
tff(fact_8537_of__int__leD,axiom,
    ! [Nb: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xc)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xc) ) ) ).

% of_int_leD
tff(fact_8538_of__int__lessD,axiom,
    ! [Nb: int,Xc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(ring_18347121197199848620nteger(Nb))),Xc)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xc) ) ) ).

% of_int_lessD
tff(fact_8539_of__int__lessD,axiom,
    ! [Nb: int,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xc)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc) ) ) ).

% of_int_lessD
tff(fact_8540_of__int__lessD,axiom,
    ! [Nb: int,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xc)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xc) ) ) ).

% of_int_lessD
tff(fact_8541_of__int__lessD,axiom,
    ! [Nb: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xc)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xc) ) ) ).

% of_int_lessD
tff(fact_8542_lemma__interval,axiom,
    ! [A2: real,Xc: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),B2)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [Y3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Y3))),D5)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_8543_round__diff__minimal,axiom,
    ! [Z: real,M: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Z))))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z),aa_int_real(ring_1_of_int_real,M)))) ).

% round_diff_minimal
tff(fact_8544_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Z))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Z),aa_int_rat(ring_1_of_int_rat,M)))) ).

% round_diff_minimal
tff(fact_8545_norm__triangle__ineq3,axiom,
    ! [A2: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A2)),aa_real_real(real_V7735802525324610683m_real,B2)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2))) ).

% norm_triangle_ineq3
tff(fact_8546_norm__triangle__ineq3,axiom,
    ! [A2: complex,B2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A2)),aa_complex_real(real_V1022390504157884413omplex,B2)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A2),B2))) ).

% norm_triangle_ineq3
tff(fact_8547_pi__less__4,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% pi_less_4
tff(fact_8548_pi__ge__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi) ).

% pi_ge_two
tff(fact_8549_pi__half__neq__two,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% pi_half_neq_two
tff(fact_8550_abs__le__square__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),aa_real_real(abs_abs_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_8551_abs__le__square__iff,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xc)),abs_abs_Code_integer(Ya))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_8552_abs__le__square__iff,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xc)),aa_rat_rat(abs_abs_rat,Ya))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_8553_abs__le__square__iff,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xc)),aa_int_int(abs_abs_int,Ya))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_8554_abs__square__eq__1,axiom,
    ! [Xc: rat] :
      ( ( aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,Xc) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_8555_abs__square__eq__1,axiom,
    ! [Xc: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,Xc) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_8556_abs__square__eq__1,axiom,
    ! [Xc: int] :
      ( ( aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,Xc) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_8557_abs__square__eq__1,axiom,
    ! [Xc: code_integer] :
      ( ( power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(Xc) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_8558_power__even__abs,axiom,
    ! [Nb: nat,A2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_rat(power_power_rat(aa_rat_rat(abs_abs_rat,A2)),Nb) = aa_nat_rat(power_power_rat(A2),Nb) ) ) ).

% power_even_abs
tff(fact_8559_power__even__abs,axiom,
    ! [Nb: nat,A2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb) ) ) ).

% power_even_abs
tff(fact_8560_power__even__abs,axiom,
    ! [Nb: nat,A2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_int(power_power_int(aa_int_int(abs_abs_int,A2)),Nb) = aa_nat_int(power_power_int(A2),Nb) ) ) ).

% power_even_abs
tff(fact_8561_power__even__abs,axiom,
    ! [Nb: nat,A2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( power_8256067586552552935nteger(abs_abs_Code_integer(A2),Nb) = power_8256067586552552935nteger(A2,Nb) ) ) ).

% power_even_abs
tff(fact_8562_pi__half__neq__zero,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% pi_half_neq_zero
tff(fact_8563_pi__half__less__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_less_two
tff(fact_8564_pi__half__le__two,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% pi_half_le_two
tff(fact_8565_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xc: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
         => aa_real_o(aa_real_fun_real_o(P,X3),aa_nat_real(aa_real_fun_nat_real(power_power_real,X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_real_o(aa_real_fun_real_o(P,aa_real_real(abs_abs_real,Xc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_8566_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xc: code_integer] :
      ( ! [X3: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),X3)
         => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,X3),power_8256067586552552935nteger(X3,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(P,abs_abs_Code_integer(Xc)),power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_8567_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xc: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),X3)
         => aa_rat_o(aa_rat_fun_rat_o(P,X3),aa_nat_rat(power_power_rat(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_rat_o(aa_rat_fun_rat_o(P,aa_rat_rat(abs_abs_rat,Xc)),aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_8568_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xc: int] :
      ( ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X3)
         => aa_int_o(aa_int_fun_int_o(P,X3),aa_nat_int(power_power_int(X3),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
     => aa_int_o(aa_int_fun_int_o(P,aa_int_int(abs_abs_int,Xc)),aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_8569_power2__le__iff__abs__le,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),Ya) ) ) ).

% power2_le_iff_abs_le
tff(fact_8570_power2__le__iff__abs__le,axiom,
    ! [Ya: code_integer,Xc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ya)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),power_8256067586552552935nteger(Ya,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xc)),Ya) ) ) ).

% power2_le_iff_abs_le
tff(fact_8571_power2__le__iff__abs__le,axiom,
    ! [Ya: rat,Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Ya)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xc)),Ya) ) ) ).

% power2_le_iff_abs_le
tff(fact_8572_power2__le__iff__abs__le,axiom,
    ! [Ya: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xc)),Ya) ) ) ).

% power2_le_iff_abs_le
tff(fact_8573_abs__square__le__1,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real) ) ).

% abs_square_le_1
tff(fact_8574_abs__square__le__1,axiom,
    ! [Xc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xc)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_8575_abs__square__le__1,axiom,
    ! [Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xc)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_8576_abs__square__le__1,axiom,
    ! [Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xc)),one_one_int) ) ).

% abs_square_le_1
tff(fact_8577_abs__square__less__1,axiom,
    ! [Xc: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(Xc,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Xc)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_8578_abs__square__less__1,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xc)),one_one_real) ) ).

% abs_square_less_1
tff(fact_8579_abs__square__less__1,axiom,
    ! [Xc: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Xc)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_8580_abs__square__less__1,axiom,
    ! [Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xc)),one_one_int) ) ).

% abs_square_less_1
tff(fact_8581_power__mono__even,axiom,
    ! [Nb: nat,A2: real,B2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A2)),aa_real_real(abs_abs_real,B2))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb)) ) ) ).

% power_mono_even
tff(fact_8582_power__mono__even,axiom,
    ! [Nb: nat,A2: code_integer,B2: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A2)),abs_abs_Code_integer(B2))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A2,Nb)),power_8256067586552552935nteger(B2,Nb)) ) ) ).

% power_mono_even
tff(fact_8583_power__mono__even,axiom,
    ! [Nb: nat,A2: rat,B2: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A2)),aa_rat_rat(abs_abs_rat,B2))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A2),Nb)),aa_nat_rat(power_power_rat(B2),Nb)) ) ) ).

% power_mono_even
tff(fact_8584_power__mono__even,axiom,
    ! [Nb: nat,A2: int,B2: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A2)),aa_int_int(abs_abs_int,B2))
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A2),Nb)),aa_nat_int(power_power_int(B2),Nb)) ) ) ).

% power_mono_even
tff(fact_8585_pi__half__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_gt_zero
tff(fact_8586_pi__half__ge__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% pi_half_ge_zero
tff(fact_8587_m2pi__less__pi,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))),pi) ).

% m2pi_less_pi
tff(fact_8588_minus__pi__half__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),zero_zero_real) ).

% minus_pi_half_less_zero
tff(fact_8589_of__int__round__abs__le,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xc))),Xc))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_8590_of__int__round__abs__le,axiom,
    ! [Xc: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xc))),Xc))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_8591_round__unique_H,axiom,
    ! [Xc: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),aa_int_real(ring_1_of_int_real,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,Xc) = Nb ) ) ).

% round_unique'
tff(fact_8592_round__unique_H,axiom,
    ! [Xc: rat,Nb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xc),aa_int_rat(ring_1_of_int_rat,Nb)))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))
     => ( aa_rat_int(archim7778729529865785530nd_rat,Xc) = Nb ) ) ).

% round_unique'
tff(fact_8593_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc))),Xc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_8594_ex__min__if__finite,axiom,
    ! [S: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( ( S != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),S)
            & ~ ? [Xa: code_integer] :
                  ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),S)
                  & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8595_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( ( S != bot_bot_set_o )
       => ? [X3: $o] :
            ( aa_set_o_o(member_o((X3)),S)
            & ~ ? [Xa: $o] :
                  ( aa_set_o_o(member_o((Xa)),S)
                  & aa_o_o(ord_less_o((Xa)),(X3)) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8596_ex__min__if__finite,axiom,
    ! [S: set_real] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( ( S != bot_bot_set_real )
       => ? [X3: real] :
            ( aa_set_real_o(member_real(X3),S)
            & ~ ? [Xa: real] :
                  ( aa_set_real_o(member_real(Xa),S)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8597_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( aa_set_rat_o(finite_finite_rat,S)
     => ( ( S != bot_bot_set_rat )
       => ? [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),S)
            & ~ ? [Xa: rat] :
                  ( aa_set_rat_o(member_rat(Xa),S)
                  & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8598_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( aa_set_num_o(finite_finite_num,S)
     => ( ( S != bot_bot_set_num )
       => ? [X3: num] :
            ( aa_set_num_o(member_num(X3),S)
            & ~ ? [Xa: num] :
                  ( aa_set_num_o(member_num(Xa),S)
                  & aa_num_o(aa_num_fun_num_o(ord_less_num,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8599_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( ( S != bot_bot_set_nat )
       => ? [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),S)
            & ~ ? [Xa: nat] :
                  ( aa_set_nat_o(member_nat(Xa),S)
                  & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8600_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( ( S != bot_bot_set_int )
       => ? [X3: int] :
            ( aa_set_int_o(member_int(X3),S)
            & ~ ? [Xa: int] :
                  ( aa_set_int_o(member_int(Xa),S)
                  & aa_int_o(aa_int_fun_int_o(ord_less_int,Xa),X3) ) ) ) ) ).

% ex_min_if_finite
tff(fact_8601_infinite__growing,axiom,
    ! [X: set_Code_integer] :
      ( ( X != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( aa_set5643387332196575111eger_o(member_Code_integer(X3),X)
           => ? [Xa: code_integer] :
                ( aa_set5643387332196575111eger_o(member_Code_integer(Xa),X)
                & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X3),Xa) ) )
       => ~ aa_set5643387332196575111eger_o(finite6017078050557962740nteger,X) ) ) ).

% infinite_growing
tff(fact_8602_infinite__growing,axiom,
    ! [X: set_o] :
      ( ( X != bot_bot_set_o )
     => ( ! [X3: $o] :
            ( aa_set_o_o(member_o((X3)),X)
           => ? [Xa: $o] :
                ( aa_set_o_o(member_o((Xa)),X)
                & aa_o_o(ord_less_o((X3)),(Xa)) ) )
       => ~ aa_set_o_o(finite_finite_o,X) ) ) ).

% infinite_growing
tff(fact_8603_infinite__growing,axiom,
    ! [X: set_real] :
      ( ( X != bot_bot_set_real )
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),X)
           => ? [Xa: real] :
                ( aa_set_real_o(member_real(Xa),X)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Xa) ) )
       => ~ aa_set_real_o(finite_finite_real,X) ) ) ).

% infinite_growing
tff(fact_8604_infinite__growing,axiom,
    ! [X: set_rat] :
      ( ( X != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( aa_set_rat_o(member_rat(X3),X)
           => ? [Xa: rat] :
                ( aa_set_rat_o(member_rat(Xa),X)
                & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Xa) ) )
       => ~ aa_set_rat_o(finite_finite_rat,X) ) ) ).

% infinite_growing
tff(fact_8605_infinite__growing,axiom,
    ! [X: set_num] :
      ( ( X != bot_bot_set_num )
     => ( ! [X3: num] :
            ( aa_set_num_o(member_num(X3),X)
           => ? [Xa: num] :
                ( aa_set_num_o(member_num(Xa),X)
                & aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Xa) ) )
       => ~ aa_set_num_o(finite_finite_num,X) ) ) ).

% infinite_growing
tff(fact_8606_infinite__growing,axiom,
    ! [X: set_nat] :
      ( ( X != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( aa_set_nat_o(member_nat(X3),X)
           => ? [Xa: nat] :
                ( aa_set_nat_o(member_nat(Xa),X)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Xa) ) )
       => ~ aa_set_nat_o(finite_finite_nat,X) ) ) ).

% infinite_growing
tff(fact_8607_infinite__growing,axiom,
    ! [X: set_int] :
      ( ( X != bot_bot_set_int )
     => ( ! [X3: int] :
            ( aa_set_int_o(member_int(X3),X)
           => ? [Xa: int] :
                ( aa_set_int_o(member_int(Xa),X)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Xa) ) )
       => ~ aa_set_int_o(finite_finite_int,X) ) ) ).

% infinite_growing
tff(fact_8608_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc))),Xc))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_8609_finite__ranking__induct,axiom,
    ! [S: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,F2: fun_VEBT_VEBT_rat] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X3: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S5)
             => ( ! [Y3: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y3),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_VEBT_VEBT_rat(F2,Y3)),aa_VEBT_VEBT_rat(F2,X3)) )
               => ( aa_set_VEBT_VEBT_o(P,S5)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X3,S5)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8610_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,P: fun_se7289222521248229150eger_o,F2: fun_Code_integer_rat] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X3: code_integer,S5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S5)
             => ( ! [Y3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y3),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Code_integer_rat(F2,Y3)),aa_Code_integer_rat(F2,X3)) )
               => ( aa_set5643387332196575111eger_o(P,S5)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(X3,S5)) ) ) )
         => aa_set5643387332196575111eger_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8611_finite__ranking__induct,axiom,
    ! [S: set_complex,P: fun_set_complex_o,F2: fun_complex_rat] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,S5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S5)
             => ( ! [Y3: complex] :
                    ( aa_set_complex_o(member_complex(Y3),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F2,Y3)),aa_complex_rat(F2,X3)) )
               => ( aa_set_complex_o(P,S5)
                 => aa_set_complex_o(P,insert_complex(X3,S5)) ) ) )
         => aa_set_complex_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8612_finite__ranking__induct,axiom,
    ! [S: set_real,P: fun_set_real_o,F2: fun_real_rat] :
      ( aa_set_real_o(finite_finite_real,S)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [X3: real,S5: set_real] :
              ( aa_set_real_o(finite_finite_real,S5)
             => ( ! [Y3: real] :
                    ( aa_set_real_o(member_real(Y3),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F2,Y3)),aa_real_rat(F2,X3)) )
               => ( aa_set_real_o(P,S5)
                 => aa_set_real_o(P,insert_real(X3,S5)) ) ) )
         => aa_set_real_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8613_finite__ranking__induct,axiom,
    ! [S: set_o,P: fun_set_o_o,F2: fun_o_rat] :
      ( aa_set_o_o(finite_finite_o,S)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [X3: $o,S5: set_o] :
              ( aa_set_o_o(finite_finite_o,S5)
             => ( ! [Y3: $o] :
                    ( aa_set_o_o(member_o((Y3)),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_o_rat(F2,(Y3))),aa_o_rat(F2,(X3))) )
               => ( aa_set_o_o(P,S5)
                 => aa_set_o_o(P,insert_o((X3),S5)) ) ) )
         => aa_set_o_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8614_finite__ranking__induct,axiom,
    ! [S: set_nat,P: fun_set_nat_o,F2: fun_nat_rat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [X3: nat,S5: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,S5)
             => ( ! [Y3: nat] :
                    ( aa_set_nat_o(member_nat(Y3),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F2,Y3)),aa_nat_rat(F2,X3)) )
               => ( aa_set_nat_o(P,S5)
                 => aa_set_nat_o(P,insert_nat(X3,S5)) ) ) )
         => aa_set_nat_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8615_finite__ranking__induct,axiom,
    ! [S: set_int,P: fun_set_int_o,F2: fun_int_rat] :
      ( aa_set_int_o(finite_finite_int,S)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [X3: int,S5: set_int] :
              ( aa_set_int_o(finite_finite_int,S5)
             => ( ! [Y3: int] :
                    ( aa_set_int_o(member_int(Y3),S5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F2,Y3)),aa_int_rat(F2,X3)) )
               => ( aa_set_int_o(P,S5)
                 => aa_set_int_o(P,insert_int(X3,S5)) ) ) )
         => aa_set_int_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8616_finite__ranking__induct,axiom,
    ! [S: set_VEBT_VEBT,P: fun_set_VEBT_VEBT_o,F2: fun_VEBT_VEBT_num] :
      ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S)
     => ( aa_set_VEBT_VEBT_o(P,bot_bo8194388402131092736T_VEBT)
       => ( ! [X3: vEBT_VEBT,S5: set_VEBT_VEBT] :
              ( aa_set_VEBT_VEBT_o(finite5795047828879050333T_VEBT,S5)
             => ( ! [Y3: vEBT_VEBT] :
                    ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Y3),S5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_VEBT_VEBT_num(F2,Y3)),aa_VEBT_VEBT_num(F2,X3)) )
               => ( aa_set_VEBT_VEBT_o(P,S5)
                 => aa_set_VEBT_VEBT_o(P,insert_VEBT_VEBT(X3,S5)) ) ) )
         => aa_set_VEBT_VEBT_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8617_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,P: fun_se7289222521248229150eger_o,F2: fun_Code_integer_num] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [X3: code_integer,S5: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,S5)
             => ( ! [Y3: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(Y3),S5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_Code_integer_num(F2,Y3)),aa_Code_integer_num(F2,X3)) )
               => ( aa_set5643387332196575111eger_o(P,S5)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(X3,S5)) ) ) )
         => aa_set5643387332196575111eger_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8618_finite__ranking__induct,axiom,
    ! [S: set_complex,P: fun_set_complex_o,F2: fun_complex_num] :
      ( aa_set_complex_o(finite3207457112153483333omplex,S)
     => ( aa_set_complex_o(P,bot_bot_set_complex)
       => ( ! [X3: complex,S5: set_complex] :
              ( aa_set_complex_o(finite3207457112153483333omplex,S5)
             => ( ! [Y3: complex] :
                    ( aa_set_complex_o(member_complex(Y3),S5)
                   => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_complex_num(F2,Y3)),aa_complex_num(F2,X3)) )
               => ( aa_set_complex_o(P,S5)
                 => aa_set_complex_o(P,insert_complex(X3,S5)) ) ) )
         => aa_set_complex_o(P,S) ) ) ) ).

% finite_ranking_induct
tff(fact_8619_finite__linorder__min__induct,axiom,
    ! [A: set_Code_integer,P: fun_se7289222521248229150eger_o] :
      ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A)
     => ( aa_set5643387332196575111eger_o(P,bot_bo3990330152332043303nteger)
       => ( ! [B3: code_integer,A7: set_Code_integer] :
              ( aa_set5643387332196575111eger_o(finite6017078050557962740nteger,A7)
             => ( ! [X4: code_integer] :
                    ( aa_set5643387332196575111eger_o(member_Code_integer(X4),A7)
                   => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B3),X4) )
               => ( aa_set5643387332196575111eger_o(P,A7)
                 => aa_set5643387332196575111eger_o(P,insert_Code_integer(B3,A7)) ) ) )
         => aa_set5643387332196575111eger_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8620_finite__linorder__min__induct,axiom,
    ! [A: set_o,P: fun_set_o_o] :
      ( aa_set_o_o(finite_finite_o,A)
     => ( aa_set_o_o(P,bot_bot_set_o)
       => ( ! [B3: $o,A7: set_o] :
              ( aa_set_o_o(finite_finite_o,A7)
             => ( ! [X4: $o] :
                    ( aa_set_o_o(member_o((X4)),A7)
                   => aa_o_o(ord_less_o((B3)),(X4)) )
               => ( aa_set_o_o(P,A7)
                 => aa_set_o_o(P,insert_o((B3),A7)) ) ) )
         => aa_set_o_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8621_finite__linorder__min__induct,axiom,
    ! [A: set_real,P: fun_set_real_o] :
      ( aa_set_real_o(finite_finite_real,A)
     => ( aa_set_real_o(P,bot_bot_set_real)
       => ( ! [B3: real,A7: set_real] :
              ( aa_set_real_o(finite_finite_real,A7)
             => ( ! [X4: real] :
                    ( aa_set_real_o(member_real(X4),A7)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,B3),X4) )
               => ( aa_set_real_o(P,A7)
                 => aa_set_real_o(P,insert_real(B3,A7)) ) ) )
         => aa_set_real_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8622_finite__linorder__min__induct,axiom,
    ! [A: set_rat,P: fun_set_rat_o] :
      ( aa_set_rat_o(finite_finite_rat,A)
     => ( aa_set_rat_o(P,bot_bot_set_rat)
       => ( ! [B3: rat,A7: set_rat] :
              ( aa_set_rat_o(finite_finite_rat,A7)
             => ( ! [X4: rat] :
                    ( aa_set_rat_o(member_rat(X4),A7)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B3),X4) )
               => ( aa_set_rat_o(P,A7)
                 => aa_set_rat_o(P,insert_rat(B3,A7)) ) ) )
         => aa_set_rat_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8623_finite__linorder__min__induct,axiom,
    ! [A: set_num,P: fun_set_num_o] :
      ( aa_set_num_o(finite_finite_num,A)
     => ( aa_set_num_o(P,bot_bot_set_num)
       => ( ! [B3: num,A7: set_num] :
              ( aa_set_num_o(finite_finite_num,A7)
             => ( ! [X4: num] :
                    ( aa_set_num_o(member_num(X4),A7)
                   => aa_num_o(aa_num_fun_num_o(ord_less_num,B3),X4) )
               => ( aa_set_num_o(P,A7)
                 => aa_set_num_o(P,insert_num(B3,A7)) ) ) )
         => aa_set_num_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8624_finite__linorder__min__induct,axiom,
    ! [A: set_nat,P: fun_set_nat_o] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => ( aa_set_nat_o(P,bot_bot_set_nat)
       => ( ! [B3: nat,A7: set_nat] :
              ( aa_set_nat_o(finite_finite_nat,A7)
             => ( ! [X4: nat] :
                    ( aa_set_nat_o(member_nat(X4),A7)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B3),X4) )
               => ( aa_set_nat_o(P,A7)
                 => aa_set_nat_o(P,insert_nat(B3,A7)) ) ) )
         => aa_set_nat_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8625_finite__linorder__min__induct,axiom,
    ! [A: set_int,P: fun_set_int_o] :
      ( aa_set_int_o(finite_finite_int,A)
     => ( aa_set_int_o(P,bot_bot_set_int)
       => ( ! [B3: int,A7: set_int] :
              ( aa_set_int_o(finite_finite_int,A7)
             => ( ! [X4: int] :
                    ( aa_set_int_o(member_int(X4),A7)
                   => aa_int_o(aa_int_fun_int_o(ord_less_int,B3),X4) )
               => ( aa_set_int_o(P,A7)
                 => aa_set_int_o(P,insert_int(B3,A7)) ) ) )
         => aa_set_int_o(P,A) ) ) ) ).

% finite_linorder_min_induct
tff(fact_8626_monoseq__arctan__series,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_hn,Xc)) ) ).

% monoseq_arctan_series
tff(fact_8627_arctan__series,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => ( aa_real_real(arctan,Xc) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ho,Xc)) ) ) ).

% arctan_series
tff(fact_8628_sin__cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% sin_cos_npi
tff(fact_8629_signed__take__bit__rec,axiom,
    ! [Nb: nat,A2: code_integer] :
      bit_ri6519982836138164636nteger(Nb,A2) = $ite(Nb = zero_zero_nat,uminus1351360451143612070nteger(modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))),plus_p5714425477246183910nteger(modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat),divide6298287555418463151nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_8630_signed__take__bit__rec,axiom,
    ! [Nb: nat,A2: int] :
      aa_int_int(bit_ri631733984087533419it_int(Nb),A2) = $ite(Nb = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_8631_dbl__simps_I4_J,axiom,
    aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_8632_dbl__simps_I4_J,axiom,
    aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_8633_dbl__simps_I4_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_8634_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_8635_dbl__simps_I4_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% dbl_simps(4)
tff(fact_8636_signed__take__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),zero_zero_int) = zero_zero_int ).

% signed_take_bit_of_0
tff(fact_8637_sin__zero,axiom,
    aa_complex_complex(sin_complex,zero_zero_complex) = zero_zero_complex ).

% sin_zero
tff(fact_8638_sin__zero,axiom,
    aa_real_real(sin_real,zero_zero_real) = zero_zero_real ).

% sin_zero
tff(fact_8639_arctan__zero__zero,axiom,
    aa_real_real(arctan,zero_zero_real) = zero_zero_real ).

% arctan_zero_zero
tff(fact_8640_arctan__eq__zero__iff,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(arctan,Xc) = zero_zero_real )
    <=> ( Xc = zero_zero_real ) ) ).

% arctan_eq_zero_iff
tff(fact_8641_dbl__simps_I2_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,zero_zero_complex) = zero_zero_complex ).

% dbl_simps(2)
tff(fact_8642_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_8643_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_8644_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_8645_signed__take__bit__Suc__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),one_one_int) = one_one_int ).

% signed_take_bit_Suc_1
tff(fact_8646_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_8647_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : bit_ri6519982836138164636nteger(Nb,uminus1351360451143612070nteger(one_one_Code_integer)) = uminus1351360451143612070nteger(one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_8648_signed__take__bit__of__minus__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% signed_take_bit_of_minus_1
tff(fact_8649_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Z)),one_one_int)
    <=> ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_8650_sin__pi,axiom,
    aa_real_real(sin_real,pi) = zero_zero_real ).

% sin_pi
tff(fact_8651_arctan__less__zero__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xc)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_8652_zero__less__arctan__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc) ) ).

% zero_less_arctan_iff
tff(fact_8653_arctan__le__zero__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xc)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_8654_zero__le__arctan__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc) ) ).

% zero_le_arctan_iff
tff(fact_8655_sin__pi__minus,axiom,
    ! [Xc: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),Xc)) = aa_real_real(sin_real,Xc) ).

% sin_pi_minus
tff(fact_8656_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K)) = numera6690914467698888265omplex(aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_8657_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K)) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_8658_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K)) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_8659_dbl__simps_I5_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)) ).

% dbl_simps(5)
tff(fact_8660_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_real_real(neg_numeral_dbl_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,K))) = aa_real_real(uminus_uminus_real,aa_real_real(neg_numeral_dbl_real,aa_num_real(numeral_numeral_real,K))) ).

% dbl_simps(1)
tff(fact_8661_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_int_int(neg_numeral_dbl_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(uminus_uminus_int,aa_int_int(neg_numeral_dbl_int,aa_num_int(numeral_numeral_int,K))) ).

% dbl_simps(1)
tff(fact_8662_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_complex_complex(neg_nu7009210354673126013omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(neg_nu7009210354673126013omplex,numera6690914467698888265omplex(K))) ).

% dbl_simps(1)
tff(fact_8663_dbl__simps_I1_J,axiom,
    ! [K: num] : neg_nu8804712462038260780nteger(uminus1351360451143612070nteger(numera6620942414471956472nteger(K))) = uminus1351360451143612070nteger(neg_nu8804712462038260780nteger(numera6620942414471956472nteger(K))) ).

% dbl_simps(1)
tff(fact_8664_dbl__simps_I1_J,axiom,
    ! [K: num] : aa_rat_rat(neg_numeral_dbl_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,K))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(neg_numeral_dbl_rat,aa_num_rat(numeral_numeral_rat,K))) ).

% dbl_simps(1)
tff(fact_8665_sin__periodic__pi2,axiom,
    ! [Xc: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xc)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xc)) ).

% sin_periodic_pi2
tff(fact_8666_sin__periodic__pi,axiom,
    ! [Xc: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xc)) ).

% sin_periodic_pi
tff(fact_8667_sin__minus__pi,axiom,
    ! [Xc: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xc)) ).

% sin_minus_pi
tff(fact_8668_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_bit0
tff(fact_8669_sin__npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% sin_npi
tff(fact_8670_sin__npi2,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = zero_zero_real ).

% sin_npi2
tff(fact_8671_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_8672_dbl__simps_I3_J,axiom,
    aa_real_real(neg_numeral_dbl_real,one_one_real) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_8673_dbl__simps_I3_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_8674_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_8675_sin__npi__int,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_npi_int
tff(fact_8676_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_8677_signed__take__bit__0,axiom,
    ! [A2: code_integer] : bit_ri6519982836138164636nteger(zero_zero_nat,A2) = uminus1351360451143612070nteger(modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_8678_signed__take__bit__0,axiom,
    ! [A2: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A2) = aa_int_int(uminus_uminus_int,aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_8679_sin__two__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = zero_zero_real ).

% sin_two_pi
tff(fact_8680_sin__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_pi_half
tff(fact_8681_sin__periodic,axiom,
    ! [Xc: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,Xc) ).

% sin_periodic
tff(fact_8682_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_8683_sin__2npi,axiom,
    ! [Nb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = zero_zero_real ).

% sin_2npi
tff(fact_8684_sin__2pi__minus,axiom,
    ! [Xc: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xc)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xc)) ).

% sin_2pi_minus
tff(fact_8685_sin__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = zero_zero_real ).

% sin_int_2pin
tff(fact_8686_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_8687_sin__3over2__pi,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sin_3over2_pi
tff(fact_8688_signed__take__bit__add,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(plus_plus_int(aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(plus_plus_int(K),L)) ).

% signed_take_bit_add
tff(fact_8689_signed__take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(minus_minus_int(aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(minus_minus_int(K),L)) ).

% signed_take_bit_diff
tff(fact_8690_signed__take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(times_times_int(K),L)) ).

% signed_take_bit_mult
tff(fact_8691_arctan__le__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xc)),aa_real_real(arctan,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ).

% arctan_le_iff
tff(fact_8692_arctan__monotone_H,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xc)),aa_real_real(arctan,Ya)) ) ).

% arctan_monotone'
tff(fact_8693_arctan__less__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xc)),aa_real_real(arctan,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ).

% arctan_less_iff
tff(fact_8694_arctan__monotone,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xc)),aa_real_real(arctan,Ya)) ) ).

% arctan_monotone
tff(fact_8695_signed__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,K)) ).

% signed_take_bit_minus
tff(fact_8696_sin__x__le__x,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xc)),Xc) ) ).

% sin_x_le_x
tff(fact_8697_sin__le__one,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xc)),one_one_real) ).

% sin_le_one
tff(fact_8698_abs__sin__x__le__abs__x,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xc))),aa_real_real(abs_abs_real,Xc)) ).

% abs_sin_x_le_abs_x
tff(fact_8699_dbl__def,axiom,
    ! [Xc: real] : aa_real_real(neg_numeral_dbl_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Xc) ).

% dbl_def
tff(fact_8700_dbl__def,axiom,
    ! [Xc: rat] : aa_rat_rat(neg_numeral_dbl_rat,Xc) = aa_rat_rat(plus_plus_rat(Xc),Xc) ).

% dbl_def
tff(fact_8701_dbl__def,axiom,
    ! [Xc: int] : aa_int_int(neg_numeral_dbl_int,Xc) = aa_int_int(plus_plus_int(Xc),Xc) ).

% dbl_def
tff(fact_8702_abs__div,axiom,
    ! [Ya: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Ya),Xc)
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xc),Ya)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,Xc)),aa_int_int(abs_abs_int,Ya)) ) ) ).

% abs_div
tff(fact_8703_sin__gt__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xc)) ) ) ).

% sin_gt_zero
tff(fact_8704_sin__x__ge__neg__x,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xc)),aa_real_real(sin_real,Xc)) ) ).

% sin_x_ge_neg_x
tff(fact_8705_sin__ge__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xc)) ) ) ).

% sin_ge_zero
tff(fact_8706_sin__ge__minus__one,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,Xc)) ).

% sin_ge_minus_one
tff(fact_8707_abs__sin__le__one,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xc))),one_one_real) ).

% abs_sin_le_one
tff(fact_8708_zabs__def,axiom,
    ! [I: int] :
      aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ).

% zabs_def
tff(fact_8709_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(modulo_modulo_int(K),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_8710_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(times_times_int(W),Z))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,W))),aa_int_nat(nat2,aa_int_int(abs_abs_int,Z))) ).

% nat_abs_mult_distrib
tff(fact_8711_sin__eq__0__pi,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),pi)
       => ( ( aa_real_real(sin_real,Xc) = zero_zero_real )
         => ( Xc = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_8712_sin__zero__pi__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xc)),pi)
     => ( ( aa_real_real(sin_real,Xc) = zero_zero_real )
      <=> ( Xc = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_8713_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(plus_plus_int(K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_8714_sin__zero__iff__int2,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(sin_real,Xc) = zero_zero_real )
    <=> ? [I2: int] : Xc = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),pi) ) ).

% sin_zero_iff_int2
tff(fact_8715_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% div_abs_eq_div_nat
tff(fact_8716_sin__gt__zero__02,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xc)) ) ) ).

% sin_gt_zero_02
tff(fact_8717_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_8718_even__signed__take__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(M),A2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A2) ) ).

% even_signed_take_bit_iff
tff(fact_8719_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),aa_int_int(abs_abs_int,L)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),L)) ) ).

% even_add_abs_iff
tff(fact_8720_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,K)),L))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(plus_plus_int(K),L)) ) ).

% even_abs_add_iff
tff(fact_8721_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,A2)),aa_nat_int(semiri1314217659103216013at_int,B2)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A2),B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) ).

% nat_abs_int_diff
tff(fact_8722_monoseq__realpow,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xc)) ) ) ).

% monoseq_realpow
tff(fact_8723_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_ri631733984087533419it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_8724_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_8725_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_8726_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_8727_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_ri631733984087533419it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_8728_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_8729_nat__intermed__int__val,axiom,
    ! [M: nat,Nb: nat,F2: fun_nat_int,K: int] :
      ( ! [I5: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I5)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F2,aa_nat_nat(suc,I5))),aa_nat_int(F2,I5)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F2,M)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F2,Nb))
           => ? [I5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),I5)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
                & ( aa_nat_int(F2,I5) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_8730_decr__lemma,axiom,
    ! [D2: int,Xc: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(minus_minus_int(Xc),aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xc),Z))),one_one_int)),D2))),Z) ) ).

% decr_lemma
tff(fact_8731_incr__lemma,axiom,
    ! [D2: int,Z: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(plus_plus_int(Xc),aa_int_int(times_times_int(aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(Xc),Z))),one_one_int)),D2))) ) ).

% incr_lemma
tff(fact_8732_arctan__ubound,axiom,
    ! [Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arctan_ubound
tff(fact_8733_arctan__one,axiom,
    aa_real_real(arctan,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% arctan_one
tff(fact_8734_sin__gt__zero2,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xc)) ) ) ).

% sin_gt_zero2
tff(fact_8735_sin__lt__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xc)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_8736_sin__30,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_30
tff(fact_8737_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(minus_minus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_8738_sin__inj__pi,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xc) = aa_real_real(sin_real,Ya) )
             => ( Xc = Ya ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_8739_sin__mono__le__eq,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xc)),aa_real_real(sin_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_8740_sin__monotone__2pi__le,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Ya)),aa_real_real(sin_real,Xc)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_8741_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_ri631733984087533419it_int(Nb),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_8742_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_8743_nat__ivt__aux,axiom,
    ! [Nb: nat,F2: fun_nat_int,K: int] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F2,aa_nat_nat(suc,I5))),aa_nat_int(F2,I5)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F2,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F2,Nb))
         => ? [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
              & ( aa_nat_int(F2,I5) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_8744_arctan__lbound,axiom,
    ! [Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Ya)) ).

% arctan_lbound
tff(fact_8745_arctan__bounded,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Ya))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% arctan_bounded
tff(fact_8746_sin__le__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xc)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_8747_sin__less__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xc)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_8748_sin__mono__less__eq,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xc)),aa_real_real(sin_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_8749_sin__monotone__2pi,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Ya)),aa_real_real(sin_real,Xc)) ) ) ) ).

% sin_monotone_2pi
tff(fact_8750_sin__total,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(sin_real,X3) = Ya )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
                  & ( aa_real_real(sin_real,Y3) = Ya ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% sin_total
tff(fact_8751_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F2: fun_nat_int,K: int] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I5),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_nat_int(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I5),one_one_nat))),aa_nat_int(F2,I5)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F2,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F2,Nb))
         => ? [I5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),Nb)
              & ( aa_nat_int(F2,I5) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_8752_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_8753_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(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,Nb)))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_8754_arctan__add,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xc)),aa_real_real(arctan,Ya)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya)))) ) ) ) ).

% arctan_add
tff(fact_8755_machin__Euler,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,one))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))))))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) ).

% machin_Euler
tff(fact_8756_machin,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit1,one))))))))))) ).

% machin
tff(fact_8757_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A2: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,Nb),A2) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one))),times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),bit_ri6519982836138164636nteger(Nb,divide6298287555418463151nteger(A2,numera6620942414471956472nteger(aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_8758_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A2: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),A2) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_8759_sin__zero__iff__int,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(sin_real,Xc) = zero_zero_real )
    <=> ? [I2: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I2)
          & ( Xc = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_8760_sin__zero__lemma,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( ( aa_real_real(sin_real,Xc) = zero_zero_real )
       => ? [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( Xc = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_8761_sin__zero__iff,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(sin_real,Xc) = zero_zero_real )
    <=> ( ? [N6: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)
            & ( Xc = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N6: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)
            & ( Xc = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_8762_arctan__double,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,Xc)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_8763_summable__arctan__series,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_ho,Xc)) ) ).

% summable_arctan_series
tff(fact_8764_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))),one_one_int))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_8765_sincos__total__2pi,axiom,
    ! [Xc: real,Ya: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( Xc = aa_real_real(cos_real,T5) )
               => ( Ya != aa_real_real(sin_real,T5) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_8766_summable__single,axiom,
    ! [I: nat,F2: fun_nat_complex] : summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hp(I),F2)) ).

% summable_single
tff(fact_8767_summable__single,axiom,
    ! [I: nat,F2: fun_nat_real] : summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_hq(I),F2)) ).

% summable_single
tff(fact_8768_summable__single,axiom,
    ! [I: nat,F2: fun_nat_nat] : summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hr(I),F2)) ).

% summable_single
tff(fact_8769_summable__single,axiom,
    ! [I: nat,F2: fun_nat_int] : summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_hs(I),F2)) ).

% summable_single
tff(fact_8770_summable__zero,axiom,
    summable_complex(aTP_Lamm_hj) ).

% summable_zero
tff(fact_8771_summable__zero,axiom,
    summable_real(aTP_Lamm_hk) ).

% summable_zero
tff(fact_8772_summable__zero,axiom,
    summable_nat(aTP_Lamm_ct) ).

% summable_zero
tff(fact_8773_summable__zero,axiom,
    summable_int(aTP_Lamm_hl) ).

% summable_zero
tff(fact_8774_summable__iff__shift,axiom,
    ! [F2: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_ht(F2),K))
    <=> summable_real(F2) ) ).

% summable_iff_shift
tff(fact_8775_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_8776_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_8777_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_8778_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,Nb) )
    <=> ( aa_num_nat(pred_numeral,K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_8779_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa_nat_nat(suc,Nb) = aa_num_nat(numeral_numeral_nat,K) )
    <=> ( Nb = aa_num_nat(pred_numeral,K) ) ) ).

% Suc_eq_numeral
tff(fact_8780_summable__cmult__iff,axiom,
    ! [C2: complex,F2: fun_nat_complex] :
      ( summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hu(C2),F2))
    <=> ( ( C2 = zero_zero_complex )
        | summable_complex(F2) ) ) ).

% summable_cmult_iff
tff(fact_8781_summable__cmult__iff,axiom,
    ! [C2: real,F2: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_hv(C2),F2))
    <=> ( ( C2 = zero_zero_real )
        | summable_real(F2) ) ) ).

% summable_cmult_iff
tff(fact_8782_summable__divide__iff,axiom,
    ! [F2: fun_nat_complex,C2: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_hw(F2),C2))
    <=> ( ( C2 = zero_zero_complex )
        | summable_complex(F2) ) ) ).

% summable_divide_iff
tff(fact_8783_summable__divide__iff,axiom,
    ! [F2: fun_nat_real,C2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_hx(F2),C2))
    <=> ( ( C2 = zero_zero_real )
        | summable_real(F2) ) ) ).

% summable_divide_iff
tff(fact_8784_summable__If__finite,axiom,
    ! [P: fun_nat_o,F2: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hy(P),F2)) ) ).

% summable_If_finite
tff(fact_8785_summable__If__finite,axiom,
    ! [P: fun_nat_o,F2: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_hz(P),F2)) ) ).

% summable_If_finite
tff(fact_8786_summable__If__finite,axiom,
    ! [P: fun_nat_o,F2: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ia(P),F2)) ) ).

% summable_If_finite
tff(fact_8787_summable__If__finite,axiom,
    ! [P: fun_nat_o,F2: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,collect_nat(P))
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ib(P),F2)) ) ).

% summable_If_finite
tff(fact_8788_summable__If__finite__set,axiom,
    ! [A: set_nat,F2: fun_nat_complex] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => summable_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ic(A),F2)) ) ).

% summable_If_finite_set
tff(fact_8789_summable__If__finite__set,axiom,
    ! [A: set_nat,F2: fun_nat_real] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_id(A),F2)) ) ).

% summable_If_finite_set
tff(fact_8790_summable__If__finite__set,axiom,
    ! [A: set_nat,F2: fun_nat_nat] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ie(A),F2)) ) ).

% summable_If_finite_set
tff(fact_8791_summable__If__finite__set,axiom,
    ! [A: set_nat,F2: fun_nat_int] :
      ( aa_set_nat_o(finite_finite_nat,A)
     => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_if(A),F2)) ) ).

% summable_If_finite_set
tff(fact_8792_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit1,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_8793_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% less_numeral_Suc
tff(fact_8794_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% less_Suc_numeral
tff(fact_8795_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(pred_numeral,K)),Nb) ) ).

% le_numeral_Suc
tff(fact_8796_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_num_nat(pred_numeral,K)) ) ).

% le_Suc_numeral
tff(fact_8797_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(pred_numeral,K)),Nb) ).

% diff_numeral_Suc
tff(fact_8798_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(pred_numeral,K)) ).

% diff_Suc_numeral
tff(fact_8799_cos__periodic__pi,axiom,
    ! [Xc: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xc)) ).

% cos_periodic_pi
tff(fact_8800_cos__periodic__pi2,axiom,
    ! [Xc: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xc)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xc)) ).

% cos_periodic_pi2
tff(fact_8801_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_num_nat(pred_numeral,K)),Nb)) ).

% max_numeral_Suc
tff(fact_8802_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),aa_num_nat(pred_numeral,K))) ).

% max_Suc_numeral
tff(fact_8803_cos__pi__minus,axiom,
    ! [Xc: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),Xc)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xc)) ).

% cos_pi_minus
tff(fact_8804_cos__minus__pi,axiom,
    ! [Xc: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xc)) ).

% cos_minus_pi
tff(fact_8805_sin__cos__squared__add3,axiom,
    ! [Xc: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xc)),aa_real_real(cos_real,Xc))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xc)),aa_real_real(sin_real,Xc))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_8806_summable__geometric__iff,axiom,
    ! [C2: real] :
      ( summable_real(aa_real_fun_nat_real(power_power_real,C2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C2)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_8807_summable__geometric__iff,axiom,
    ! [C2: complex] :
      ( summable_complex(power_power_complex(C2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C2)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_8808_cos__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_half
tff(fact_8809_cos__two__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_real ).

% cos_two_pi
tff(fact_8810_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_8811_cos__periodic,axiom,
    ! [Xc: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,Xc) ).

% cos_periodic
tff(fact_8812_cos__2pi__minus,axiom,
    ! [Xc: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xc)) = aa_real_real(cos_real,Xc) ).

% cos_2pi_minus
tff(fact_8813_cos__npi2,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi2
tff(fact_8814_cos__npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb) ).

% cos_npi
tff(fact_8815_sin__cos__squared__add2,axiom,
    ! [Xc: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_8816_sin__cos__squared__add2,axiom,
    ! [Xc: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_8817_sin__cos__squared__add,axiom,
    ! [Xc: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_8818_sin__cos__squared__add,axiom,
    ! [Xc: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_8819_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_8820_cos__2npi,axiom,
    ! [Nb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)) = one_one_real ).

% cos_2npi
tff(fact_8821_cos__int__2pin,axiom,
    ! [Nb: int] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_int_real(ring_1_of_int_real,Nb))) = one_one_real ).

% cos_int_2pin
tff(fact_8822_cos__3over2__pi,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) = zero_zero_real ).

% cos_3over2_pi
tff(fact_8823_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit1,K))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_8824_cos__npi__int,axiom,
    ! [Nb: int] :
      aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_int_real(ring_1_of_int_real,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ).

% cos_npi_int
tff(fact_8825_cos__pi__eq__zero,axiom,
    ! [M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_8826_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N5: nat,F2: fun_nat_real] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F2,N))),aa_nat_real(G,N)) )
       => summable_real(F2) ) ) ).

% summable_comparison_test'
tff(fact_8827_summable__comparison__test_H,axiom,
    ! [G: fun_nat_real,N5: nat,F2: fun_nat_complex] :
      ( summable_real(G)
     => ( ! [N: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N5),N)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F2,N))),aa_nat_real(G,N)) )
       => summable_complex(F2) ) ) ).

% summable_comparison_test'
tff(fact_8828_summable__comparison__test,axiom,
    ! [F2: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(F2,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(F2) ) ) ).

% summable_comparison_test
tff(fact_8829_summable__comparison__test,axiom,
    ! [F2: fun_nat_complex,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(F2,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_complex(F2) ) ) ).

% summable_comparison_test
tff(fact_8830_summable__const__iff,axiom,
    ! [C2: complex] :
      ( summable_complex(aTP_Lamm_ig(C2))
    <=> ( C2 = zero_zero_complex ) ) ).

% summable_const_iff
tff(fact_8831_summable__const__iff,axiom,
    ! [C2: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_ih,C2))
    <=> ( C2 = zero_zero_real ) ) ).

% summable_const_iff
tff(fact_8832_summable__add,axiom,
    ! [F2: fun_nat_real,G: fun_nat_real] :
      ( summable_real(F2)
     => ( summable_real(G)
       => summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_ii(F2),G)) ) ) ).

% summable_add
tff(fact_8833_summable__add,axiom,
    ! [F2: fun_nat_nat,G: fun_nat_nat] :
      ( summable_nat(F2)
     => ( summable_nat(G)
       => summable_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ij(F2),G)) ) ) ).

% summable_add
tff(fact_8834_summable__add,axiom,
    ! [F2: fun_nat_int,G: fun_nat_int] :
      ( summable_int(F2)
     => ( summable_int(G)
       => summable_int(aa_fun5565043122120104496at_int(aTP_Lamm_ik(F2),G)) ) ) ).

% summable_add
tff(fact_8835_summable__mult2,axiom,
    ! [F2: fun_nat_real,C2: real] :
      ( summable_real(F2)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_il(F2),C2)) ) ).

% summable_mult2
tff(fact_8836_summable__mult,axiom,
    ! [F2: fun_nat_real,C2: real] :
      ( summable_real(F2)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_im(F2),C2)) ) ).

% summable_mult
tff(fact_8837_summable__Suc__iff,axiom,
    ! [F2: fun_nat_real] :
      ( summable_real(aTP_Lamm_in(F2))
    <=> summable_real(F2) ) ).

% summable_Suc_iff
tff(fact_8838_cos__le__one,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xc)),one_one_real) ).

% cos_le_one
tff(fact_8839_polar__Ex,axiom,
    ! [Xc: real,Ya: real] :
    ? [R2: real,A3: real] :
      ( ( Xc = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(cos_real,A3)) )
      & ( Ya = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(sin_real,A3)) ) ) ).

% polar_Ex
tff(fact_8840_cos__arctan__not__zero,axiom,
    ! [Xc: real] : aa_real_real(cos_real,aa_real_real(arctan,Xc)) != zero_zero_real ).

% cos_arctan_not_zero
tff(fact_8841_numeral__eq__Suc,axiom,
    ! [K: num] : aa_num_nat(numeral_numeral_nat,K) = aa_nat_nat(suc,aa_num_nat(pred_numeral,K)) ).

% numeral_eq_Suc
tff(fact_8842_cos__inj__pi,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),pi)
           => ( ( aa_real_real(cos_real,Xc) = aa_real_real(cos_real,Ya) )
             => ( Xc = Ya ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_8843_cos__mono__le__eq,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xc)),aa_real_real(cos_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_8844_cos__monotone__0__pi__le,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xc)),aa_real_real(cos_real,Ya)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_8845_cos__ge__minus__one,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,Xc)) ).

% cos_ge_minus_one
tff(fact_8846_abs__cos__le__one,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,Xc))),one_one_real) ).

% abs_cos_le_one
tff(fact_8847_summable__rabs__comparison__test,axiom,
    ! [F2: fun_nat_real,G: fun_nat_real] :
      ( ? [N7: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F2,N))),aa_nat_real(G,N)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_io(F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_8848_pred__numeral__def,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,K)),one_one_nat) ).

% pred_numeral_def
tff(fact_8849_summable__rabs,axiom,
    ! [F2: fun_nat_real] :
      ( summable_real(aTP_Lamm_io(F2))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F2))),suminf_real(aTP_Lamm_io(F2))) ) ).

% summable_rabs
tff(fact_8850_cos__two__neq__zero,axiom,
    aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) != zero_zero_real ).

% cos_two_neq_zero
tff(fact_8851_cos__monotone__0__pi,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xc)),aa_real_real(cos_real,Ya)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_8852_cos__mono__less__eq,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xc)),aa_real_real(cos_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_8853_cos__monotone__minus__pi__0_H,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Ya)),aa_real_real(cos_real,Xc)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_8854_sin__zero__abs__cos__one,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(sin_real,Xc) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,Xc)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_8855_cos__two__less__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_less_zero
tff(fact_8856_cos__is__zero,axiom,
    ? [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
      & ( aa_real_real(cos_real,X3) = zero_zero_real )
      & ! [Y3: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
            & ( aa_real_real(cos_real,Y3) = zero_zero_real ) )
         => ( Y3 = X3 ) ) ) ).

% cos_is_zero
tff(fact_8857_cos__two__le__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),zero_zero_real) ).

% cos_two_le_zero
tff(fact_8858_cos__monotone__minus__pi__0,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Ya)),aa_real_real(cos_real,Xc)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_8859_cos__total,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => ? [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),pi)
            & ( aa_real_real(cos_real,X3) = Ya )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
                  & ( aa_real_real(cos_real,Y3) = Ya ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% cos_total
tff(fact_8860_sincos__principal__value,axiom,
    ! [Xc: real] :
    ? [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
      & ( aa_real_real(sin_real,Y) = aa_real_real(sin_real,Xc) )
      & ( aa_real_real(cos_real,Y) = aa_real_real(cos_real,Xc) ) ) ).

% sincos_principal_value
tff(fact_8861_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(M),aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_8862_summable__power__series,axiom,
    ! [F2: fun_nat_real,Z: real] :
      ( ! [I5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F2,I5)),one_one_real)
     => ( ! [I5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F2,I5))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Z)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_ip(F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_8863_sin__cos__le1,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xc)),aa_real_real(sin_real,Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xc)),aa_real_real(cos_real,Ya))))),one_one_real) ).

% sin_cos_le1
tff(fact_8864_cos__double__less__one,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xc))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_8865_cos__gt__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xc)) ) ) ).

% cos_gt_zero
tff(fact_8866_cos__60,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_60
tff(fact_8867_cos__one__2pi__int,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(cos_real,Xc) = one_one_real )
    <=> ? [X2: int] : Xc = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_8868_cos__gt__zero__pi,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xc)) ) ) ).

% cos_gt_zero_pi
tff(fact_8869_cos__ge__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,Xc)) ) ) ).

% cos_ge_zero
tff(fact_8870_cos__one__2pi,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(cos_real,Xc) = one_one_real )
    <=> ( ? [X2: nat] : Xc = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X2: nat] : Xc = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_8871_sincos__total__pi,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),pi)
            & ( Xc = aa_real_real(cos_real,T5) )
            & ( Ya = aa_real_real(sin_real,T5) ) ) ) ) ).

% sincos_total_pi
tff(fact_8872_sin__expansion__lemma,axiom,
    ! [Xc: real,M: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_8873_cos__zero__iff__int,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(cos_real,Xc) = zero_zero_real )
    <=> ? [I2: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I2)
          & ( Xc = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_8874_cos__zero__lemma,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( ( aa_real_real(cos_real,Xc) = zero_zero_real )
       => ? [N: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N)
            & ( Xc = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_8875_cos__zero__iff,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(cos_real,Xc) = zero_zero_real )
    <=> ( ? [N6: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)
            & ( Xc = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) )
        | ? [N6: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N6)
            & ( Xc = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N6)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_8876_cos__expansion__lemma,axiom,
    ! [Xc: real,M: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,M))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_8877_sincos__total__pi__half,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T5: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( Xc = aa_real_real(cos_real,T5) )
              & ( Ya = aa_real_real(sin_real,T5) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_8878_sincos__total__2pi__le,axiom,
    ! [Xc: real,Ya: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( Xc = aa_real_real(cos_real,T5) )
          & ( Ya = aa_real_real(sin_real,T5) ) ) ) ).

% sincos_total_2pi_le
tff(fact_8879_infinite__int__iff__unbounded,axiom,
    ! [S: set_int] :
      ( ~ aa_set_int_o(finite_finite_int,S)
    <=> ! [M8: int] :
        ? [N6: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M8),aa_int_int(abs_abs_int,N6))
          & aa_set_int_o(member_int(N6),S) ) ) ).

% infinite_int_iff_unbounded
tff(fact_8880_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Z) = one_one_real )
     => ~ ! [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( Z != complex2(aa_real_real(cos_real,T5),aa_real_real(sin_real,T5)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_8881_vebt__buildup_Oelims,axiom,
    ! [Xc: nat,Ya: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Xc) = Ya )
     => ( ( ( Xc = zero_zero_nat )
         => ( Ya != aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) ) )
       => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
           => ( Ya != aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
               => ( Ya != $ite(
                      aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half)) ),
                      $let(
                        half: nat,
                        half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_8882_tan__pi,axiom,
    aa_real_real(tan_real,pi) = zero_zero_real ).

% tan_pi
tff(fact_8883_tan__periodic__pi,axiom,
    ! [Xc: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),pi)) = aa_real_real(tan_real,Xc) ).

% tan_periodic_pi
tff(fact_8884_tan__npi,axiom,
    ! [Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% tan_npi
tff(fact_8885_tan__periodic__n,axiom,
    ! [Xc: real,Nb: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Nb)),pi))) = aa_real_real(tan_real,Xc) ).

% tan_periodic_n
tff(fact_8886_tan__periodic__nat,axiom,
    ! [Xc: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,Xc) ).

% tan_periodic_nat
tff(fact_8887_tan__periodic__int,axiom,
    ! [Xc: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,Xc) ).

% tan_periodic_int
tff(fact_8888_tan__periodic,axiom,
    ! [Xc: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,Xc) ).

% tan_periodic
tff(fact_8889_complex__diff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa_complex_complex(minus_minus_complex(complex2(A2,B2)),complex2(C2,D2)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D2)) ).

% complex_diff
tff(fact_8890_Complex__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = zero_zero_complex )
    <=> ( ( A2 = zero_zero_real )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_0
tff(fact_8891_zero__complex_Ocode,axiom,
    zero_zero_complex = complex2(zero_zero_real,zero_zero_real) ).

% zero_complex.code
tff(fact_8892_one__complex_Ocode,axiom,
    one_one_complex = complex2(one_one_real,zero_zero_real) ).

% one_complex.code
tff(fact_8893_Complex__eq__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = one_one_complex )
    <=> ( ( A2 = one_one_real )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_8894_Complex__eq__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = numera6690914467698888265omplex(W) )
    <=> ( ( A2 = aa_num_real(numeral_numeral_real,W) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_8895_complex__add,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa_complex_complex(plus_plus_complex(complex2(A2,B2)),complex2(C2,D2)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2)) ).

% complex_add
tff(fact_8896_Complex__eq__neg__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A2 = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_8897_Complex__eq__neg__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( A2 = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_8898_complex__mult,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa_complex_complex(times_times_complex(complex2(A2,B2)),complex2(C2,D2)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),C2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),D2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),D2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C2))) ).

% complex_mult
tff(fact_8899_tan__45,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = one_one_real ).

% tan_45
tff(fact_8900_tan__gt__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,Xc)) ) ) ).

% tan_gt_zero
tff(fact_8901_lemma__tan__total,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(tan_real,X3)) ) ) ).

% lemma_tan_total
tff(fact_8902_tan__total,axiom,
    ! [Ya: real] :
    ? [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X3) = Ya )
      & ! [Y3: real] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y3)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
            & ( aa_real_real(tan_real,Y3) = Ya ) )
         => ( Y3 = X3 ) ) ) ).

% tan_total
tff(fact_8903_tan__monotone,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Ya)),aa_real_real(tan_real,Xc)) ) ) ) ).

% tan_monotone
tff(fact_8904_tan__monotone_H,axiom,
    ! [Ya: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Ya)),aa_real_real(tan_real,Xc)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_8905_tan__mono__lt__eq,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xc)),aa_real_real(tan_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_8906_lemma__tan__total1,axiom,
    ! [Ya: real] :
    ? [X3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),X3)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,X3) = Ya ) ) ).

% lemma_tan_total1
tff(fact_8907_tan__minus__45,axiom,
    aa_real_real(tan_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))) = aa_real_real(uminus_uminus_real,one_one_real) ).

% tan_minus_45
tff(fact_8908_tan__inverse,axiom,
    ! [Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(tan_real,Ya)) = aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Ya)) ).

% tan_inverse
tff(fact_8909_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
    <=> ! [M8: nat] :
        ? [N6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M8),N6)
          & aa_set_nat_o(member_nat(N6),S) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_8910_unbounded__k__infinite,axiom,
    ! [K: nat,S: set_nat] :
      ( ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M4)
         => ? [N8: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N8)
              & aa_set_nat_o(member_nat(N8),S) ) )
     => ~ aa_set_nat_o(finite_finite_nat,S) ) ).

% unbounded_k_infinite
tff(fact_8911_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ~ aa_set_nat_o(finite_finite_nat,S)
    <=> ! [M8: nat] :
        ? [N6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N6)
          & aa_set_nat_o(member_nat(N6),S) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_8912_tan__total__pos,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(tan_real,X3) = Ya ) ) ) ).

% tan_total_pos
tff(fact_8913_tan__pos__pi2__le,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,Xc)) ) ) ).

% tan_pos_pi2_le
tff(fact_8914_tan__less__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xc)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_8915_tan__mono__le,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xc)),aa_real_real(tan_real,Ya)) ) ) ) ).

% tan_mono_le
tff(fact_8916_tan__mono__le__eq,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xc)),aa_real_real(tan_real,Ya))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_8917_tan__bound__pi2,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,Xc))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_8918_arctan__unique,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,Xc) = Ya )
         => ( aa_real_real(arctan,Ya) = Xc ) ) ) ) ).

% arctan_unique
tff(fact_8919_arctan__tan,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,Xc)) = Xc ) ) ) ).

% arctan_tan
tff(fact_8920_arctan,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arctan,Ya))
      & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
      & ( aa_real_real(tan_real,aa_real_real(arctan,Ya)) = Ya ) ) ).

% arctan
tff(fact_8921_tan__total__pi4,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => ? [Z2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))),Z2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z2) = Xc ) ) ) ).

% tan_total_pi4
tff(fact_8922_vebt__buildup_Osimps_I3_J,axiom,
    ! [Vaa: nat] :
      aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half)) ),
        $let(
          half: nat,
          half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
          aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Vaa)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_8923_ceiling__log__eq__powr__iff,axiom,
    ! [Xc: real,B2: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)) = aa_int_int(plus_plus_int(aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_nat_real(semiri5074537144036343181t_real,K))),Xc)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),one_one_nat)))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_8924_powr__gt__zero,axiom,
    ! [Xc: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2))
    <=> ( Xc != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_8925_powr__nonneg__iff,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),Xc)),zero_zero_real)
    <=> ( A2 = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_8926_powr__less__cancel__iff,axiom,
    ! [Xc: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_8927_powr__eq__one__iff,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A2)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),Xc) = one_one_real )
      <=> ( Xc = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_8928_powr__one__gt__zero__iff,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),one_one_real) = Xc )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc) ) ).

% powr_one_gt_zero_iff
tff(fact_8929_powr__one,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),one_one_real) = Xc ) ) ).

% powr_one
tff(fact_8930_powr__le__cancel__iff,axiom,
    ! [Xc: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_8931_numeral__powr__numeral__real,axiom,
    ! [M: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_num_real(numeral_numeral_real,M)),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,M)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_powr_numeral_real
tff(fact_8932_log__powr__cancel,axiom,
    ! [A2: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),Ya)) = Ya ) ) ) ).

% log_powr_cancel
tff(fact_8933_powr__log__cancel,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)) = Xc ) ) ) ) ).

% powr_log_cancel
tff(fact_8934_powr__numeral,axiom,
    ! [Xc: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).

% powr_numeral
tff(fact_8935_square__powr__half,axiom,
    ! [Xc: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xc) ).

% square_powr_half
tff(fact_8936_powr__powr,axiom,
    ! [Xc: real,A2: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),B2) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),B2)) ).

% powr_powr
tff(fact_8937_powr__less__mono2__neg,axiom,
    ! [A2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_8938_powr__non__neg,axiom,
    ! [A2: real,Xc: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),Xc)),zero_zero_real) ).

% powr_non_neg
tff(fact_8939_powr__ge__pzero,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),Ya)) ).

% powr_ge_pzero
tff(fact_8940_powr__mono2,axiom,
    ! [A2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A2)) ) ) ) ).

% powr_mono2
tff(fact_8941_powr__less__mono,axiom,
    ! [A2: real,B2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),B2)) ) ) ).

% powr_less_mono
tff(fact_8942_powr__less__cancel,axiom,
    ! [Xc: real,A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2) ) ) ).

% powr_less_cancel
tff(fact_8943_powr__mono,axiom,
    ! [A2: real,B2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),B2)) ) ) ).

% powr_mono
tff(fact_8944_powr__less__mono2,axiom,
    ! [A2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A2)) ) ) ) ).

% powr_less_mono2
tff(fact_8945_powr__mono2_H,axiom,
    ! [A2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)) ) ) ) ).

% powr_mono2'
tff(fact_8946_gr__one__powr,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),Ya)) ) ) ).

% gr_one_powr
tff(fact_8947_powr__inj,axiom,
    ! [A2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),Xc) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),Ya) )
        <=> ( Xc = Ya ) ) ) ) ).

% powr_inj
tff(fact_8948_ge__one__powr__ge__zero,axiom,
    ! [Xc: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_8949_powr__mono__both,axiom,
    ! [A2: real,B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_8950_powr__le1,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_8951_powr__divide,axiom,
    ! [Xc: real,Ya: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)),A2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A2)) ) ) ) ).

% powr_divide
tff(fact_8952_powr__mult,axiom,
    ! [Xc: real,Ya: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya)),A2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A2)) ) ) ) ).

% powr_mult
tff(fact_8953_divide__powr__uminus,axiom,
    ! [A2: real,B2: real,C2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),C2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_real_real(uminus_uminus_real,C2))) ).

% divide_powr_uminus
tff(fact_8954_log__base__powr,axiom,
    ! [A2: real,B2: real,Xc: real] :
      ( ( A2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),B2)),Xc) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)),B2) ) ) ).

% log_base_powr
tff(fact_8955_log__powr,axiom,
    ! [Xc: real,B2: real,Ya: real] :
      ( ( Xc != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)) ) ) ).

% log_powr
tff(fact_8956_ln__powr,axiom,
    ! [Xc: real,Ya: real] :
      ( ( Xc != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),aa_real_real(ln_ln_real,Xc)) ) ) ).

% ln_powr
tff(fact_8957_powr__realpow,axiom,
    ! [Xc: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb) ) ) ).

% powr_realpow
tff(fact_8958_less__log__iff,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)),Xc) ) ) ) ).

% less_log_iff
tff(fact_8959_log__less__iff,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)),Ya)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)) ) ) ) ).

% log_less_iff
tff(fact_8960_less__powr__iff,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)),Ya) ) ) ) ).

% less_powr_iff
tff(fact_8961_powr__less__iff,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)),Xc)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)) ) ) ) ).

% powr_less_iff
tff(fact_8962_powr__neg__one,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xc) ) ) ).

% powr_neg_one
tff(fact_8963_powr__mult__base,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Ya)) ) ) ).

% powr_mult_base
tff(fact_8964_powr__le__iff,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)),Xc)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)) ) ) ) ).

% powr_le_iff
tff(fact_8965_le__powr__iff,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)),Ya) ) ) ) ).

% le_powr_iff
tff(fact_8966_log__le__iff,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)),Ya)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)) ) ) ) ).

% log_le_iff
tff(fact_8967_le__log__iff,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)),Xc) ) ) ) ).

% le_log_iff
tff(fact_8968_set__encode__def,axiom,
    nat_set_encode = groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% set_encode_def
tff(fact_8969_ln__powr__bound,axiom,
    ! [Xc: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),A2)),A2)) ) ) ).

% ln_powr_bound
tff(fact_8970_ln__powr__bound2,axiom,
    ! [Xc: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,Xc)),A2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A2),A2)),Xc)) ) ) ).

% ln_powr_bound2
tff(fact_8971_log__add__eq__powr,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_8972_add__log__eq__powr,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)),Xc)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_8973_minus__log__eq__powr,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Ya)),Xc)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_8974_log__minus__eq__powr,axiom,
    ! [B2: real,Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)),Ya) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_real_real(uminus_uminus_real,Ya)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_8975_powr__neg__numeral,axiom,
    ! [Xc: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,Nb))) ) ) ).

% powr_neg_numeral
tff(fact_8976_mask__eq__sum__exp__nat,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_nat_nat(suc,zero_zero_nat)) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ft,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_8977_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cn),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Nb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% gauss_sum_nat
tff(fact_8978_sum__power2,axiom,
    ! [K: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),one_one_nat) ).

% sum_power2
tff(fact_8979_Sum__Ico__nat,axiom,
    ! [M: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cn),aa_nat_set_nat(set_or4665077453230672383an_nat(M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_8980_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_iq(A2),D2)),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),D2)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% arith_series_nat
tff(fact_8981_Sum__Icc__nat,axiom,
    ! [M: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cn),aa_nat_set_nat(set_or1269000886237332187st_nat(M),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_8982_powr__int,axiom,
    ! [Xc: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),aa_int_real(ring_1_of_int_real,I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_int_nat(nat2,I)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_8983_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A2: fun_nat_nat,B2: fun_nat_nat] :
      ( ! [I5: nat,J3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),J3)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A2,I5)),aa_nat_nat(A2,J3)) ) )
     => ( ! [I5: nat,J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I5),J3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B2,J3)),aa_nat_nat(B2,I5)) ) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_set_nat_nat(groups3542108847815614940at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ir(A2),B2)),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(A2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))),aa_set_nat_nat(groups3542108847815614940at_nat(B2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_8984_sin__tan,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,Xc)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_8985_real__sqrt__eq__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( ( aa_real_real(sqrt,Xc) = aa_real_real(sqrt,Ya) )
    <=> ( Xc = Ya ) ) ).

% real_sqrt_eq_iff
tff(fact_8986_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(sqrt,Xc) = zero_zero_real )
    <=> ( Xc = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_8987_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_8988_real__sqrt__less__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xc)),aa_real_real(sqrt,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ).

% real_sqrt_less_iff
tff(fact_8989_real__sqrt__le__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xc)),aa_real_real(sqrt,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ).

% real_sqrt_le_iff
tff(fact_8990_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_8991_real__sqrt__eq__1__iff,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(sqrt,Xc) = one_one_real )
    <=> ( Xc = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_8992_finite__lessThan,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% finite_lessThan
tff(fact_8993_real__sqrt__gt__0__iff,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya) ) ).

% real_sqrt_gt_0_iff
tff(fact_8994_real__sqrt__lt__0__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xc)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_8995_real__sqrt__ge__0__iff,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya) ) ).

% real_sqrt_ge_0_iff
tff(fact_8996_real__sqrt__le__0__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xc)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_8997_real__sqrt__lt__1__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xc)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_8998_real__sqrt__gt__1__iff,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ya) ) ).

% real_sqrt_gt_1_iff
tff(fact_8999_real__sqrt__ge__1__iff,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Ya) ) ).

% real_sqrt_ge_1_iff
tff(fact_9000_real__sqrt__le__1__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xc)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_9001_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_9002_real__sqrt__mult__self,axiom,
    ! [A2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,A2)),aa_real_real(sqrt,A2)) = aa_real_real(abs_abs_real,A2) ).

% real_sqrt_mult_self
tff(fact_9003_real__sqrt__abs2,axiom,
    ! [Xc: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc)) = aa_real_real(abs_abs_real,Xc) ).

% real_sqrt_abs2
tff(fact_9004_real__sqrt__four,axiom,
    aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% real_sqrt_four
tff(fact_9005_real__sqrt__abs,axiom,
    ! [Xc: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xc) ).

% real_sqrt_abs
tff(fact_9006_real__sqrt__pow2,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xc ) ) ).

% real_sqrt_pow2
tff(fact_9007_real__sqrt__pow2__iff,axiom,
    ! [Xc: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xc )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc) ) ).

% real_sqrt_pow2_iff
tff(fact_9008_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xc: real,Ya: real,Xaa: real,Yaa: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_9009_real__sqrt__minus,axiom,
    ! [Xc: real] : aa_real_real(sqrt,aa_real_real(uminus_uminus_real,Xc)) = aa_real_real(uminus_uminus_real,aa_real_real(sqrt,Xc)) ).

% real_sqrt_minus
tff(fact_9010_real__sqrt__mult,axiom,
    ! [Xc: real,Ya: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,Xc)),aa_real_real(sqrt,Ya)) ).

% real_sqrt_mult
tff(fact_9011_real__sqrt__le__mono,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xc)),aa_real_real(sqrt,Ya)) ) ).

% real_sqrt_le_mono
tff(fact_9012_real__sqrt__power,axiom,
    ! [Xc: real,K: nat] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xc)),K) ).

% real_sqrt_power
tff(fact_9013_real__sqrt__divide,axiom,
    ! [Xc: real,Ya: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xc)),aa_real_real(sqrt,Ya)) ).

% real_sqrt_divide
tff(fact_9014_real__sqrt__less__mono,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xc)),aa_real_real(sqrt,Ya)) ) ).

% real_sqrt_less_mono
tff(fact_9015_real__sqrt__gt__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xc)) ) ).

% real_sqrt_gt_zero
tff(fact_9016_real__sqrt__ge__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Xc)) ) ).

% real_sqrt_ge_zero
tff(fact_9017_real__sqrt__eq__zero__cancel,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( ( aa_real_real(sqrt,Xc) = zero_zero_real )
       => ( Xc = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_9018_real__sqrt__ge__one,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Xc)) ) ).

% real_sqrt_ge_one
tff(fact_9019_lessThan__atLeast0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,Nb) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb) ).

% lessThan_atLeast0
tff(fact_9020_lessThan__empty__iff,axiom,
    ! [Nb: nat] :
      ( ( aa_nat_set_nat(set_ord_lessThan_nat,Nb) = bot_bot_set_nat )
    <=> ( Nb = zero_zero_nat ) ) ).

% lessThan_empty_iff
tff(fact_9021_lessThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = insert_nat(K,aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% lessThan_Suc
tff(fact_9022_real__div__sqrt,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),aa_real_real(sqrt,Xc)) = aa_real_real(sqrt,Xc) ) ) ).

% real_div_sqrt
tff(fact_9023_sqrt__add__le__add__sqrt,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xc)),aa_real_real(sqrt,Ya))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_9024_le__real__sqrt__sumsq,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),Ya)))) ).

% le_real_sqrt_sumsq
tff(fact_9025_lessThan__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_ord_lessThan_nat,aa_num_nat(pred_numeral,K))) ).

% lessThan_nat_numeral
tff(fact_9026_sqrt2__less__2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sqrt2_less_2
tff(fact_9027_sum__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_is,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_gb(Nb),C2))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_9028_real__less__rsqrt,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(sqrt,Ya)) ) ).

% real_less_rsqrt
tff(fact_9029_sqrt__le__D,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xc)),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_9030_real__le__rsqrt,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(sqrt,Ya)) ) ).

% real_le_rsqrt
tff(fact_9031_real__le__lsqrt,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xc)),Ya) ) ) ) ).

% real_le_lsqrt
tff(fact_9032_real__sqrt__unique,axiom,
    ! [Ya: real,Xc: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xc )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_real(sqrt,Xc) = Ya ) ) ) ).

% real_sqrt_unique
tff(fact_9033_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),U)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_9034_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xc: real,Ya: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Xc )
     => ( Ya = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_9035_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xc: real,Ya: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Ya )
     => ( Xc = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_9036_real__sqrt__sum__squares__ge1,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_9037_real__sqrt__sum__squares__ge2,axiom,
    ! [Ya: real,Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_9038_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),C2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,C2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,D2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_9039_sqrt__ge__absD,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),aa_real_real(sqrt,Ya))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Ya) ) ).

% sqrt_ge_absD
tff(fact_9040_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_is,collect_complex(aTP_Lamm_fr(Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_9041_cos__45,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_45
tff(fact_9042_sin__45,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_45
tff(fact_9043_tan__60,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% tan_60
tff(fact_9044_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_lessThan_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_lessThan_eq_remove0
tff(fact_9045_real__less__lsqrt,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xc)),Ya) ) ) ) ).

% real_less_lsqrt
tff(fact_9046_sqrt__sum__squares__le__sum,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_9047_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% sqrt_even_pow2
tff(fact_9048_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Xc)),aa_real_real(abs_abs_real,Ya))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_9049_real__sqrt__ge__abs2,axiom,
    ! [Ya: real,Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_9050_real__sqrt__ge__abs1,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_9051_ln__sqrt,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xc)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_9052_arsinh__real__def,axiom,
    ! [Xc: real] : aa_real_real(arsinh_real,Xc) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_9053_cos__30,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_30
tff(fact_9054_sin__60,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_60
tff(fact_9055_complex__norm,axiom,
    ! [Xc: real,Ya: real] : aa_complex_real(real_V1022390504157884413omplex,complex2(Xc,Ya)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_9056_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xc)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_9057_arsinh__real__aux,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_9058_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xc: real,Ya: real,Xaa: real,Yaa: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Yaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_9059_arith__geo__mean__sqrt,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_9060_powr__half__sqrt,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xc) ) ) ).

% powr_half_sqrt
tff(fact_9061_tan__30,axiom,
    aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,one))))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one)))) ).

% tan_30
tff(fact_9062_sum__split__even__odd,axiom,
    ! [F2: fun_nat_real,G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_it(F2),G),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aTP_Lamm_iu(F2),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),groups6591440286371151544t_real(aTP_Lamm_iv(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_9063_cos__x__y__le__one,axiom,
    ! [Xc: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_9064_real__sqrt__sum__squares__less,axiom,
    ! [Xc: real,U: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_9065_arcosh__real__def,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc)
     => ( aa_real_real(arcosh_real,Xc) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_9066_cos__arctan,axiom,
    ! [Xc: real] : aa_real_real(cos_real,aa_real_real(arctan,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_9067_sin__arctan,axiom,
    ! [Xc: real] : aa_real_real(sin_real,aa_real_real(arctan,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_9068_sqrt__sum__squares__half__less,axiom,
    ! [Xc: real,U: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_9069_sin__cos__sqrt,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xc))
     => ( aa_real_real(sin_real,Xc) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_9070_arctan__half,axiom,
    ! [Xc: real] : aa_real_real(arctan,Xc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_9071_Sum__Icc__int,axiom,
    ! [M: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_co,set_or1266510415728281911st_int(M,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(minus_minus_int(aa_int_int(times_times_int(Nb),aa_int_int(plus_plus_int(Nb),one_one_int))),aa_int_int(times_times_int(M),aa_int_int(minus_minus_int(M),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_9072_sum__pos__lt__pair,axiom,
    ! [F2: fun_nat_real,K: nat] :
      ( summable_real(F2)
     => ( ! [D5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D5)))),aa_nat_real(F2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D5)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F2,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F2)) ) ) ).

% sum_pos_lt_pair
tff(fact_9073_cos__tan,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_9074_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : groups6591440286371151544t_real(aTP_Lamm_iw,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_9075_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_9076_complex__of__real__def,axiom,
    ! [R3: real] : real_V4546457046886955230omplex(R3) = complex2(R3,zero_zero_real) ).

% complex_of_real_def
tff(fact_9077_complex__of__real__code,axiom,
    ! [X4: real] : real_V4546457046886955230omplex(X4) = complex2(X4,zero_zero_real) ).

% complex_of_real_code
tff(fact_9078_complex__eq__cancel__iff2,axiom,
    ! [Xc: real,Ya: real,Xaa: real] :
      ( ( complex2(Xc,Ya) = real_V4546457046886955230omplex(Xaa) )
    <=> ( ( Xc = Xaa )
        & ( Ya = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_9079_Complex__mult__complex__of__real,axiom,
    ! [Xc: real,Ya: real,R3: real] : aa_complex_complex(times_times_complex(complex2(Xc,Ya)),real_V4546457046886955230omplex(R3)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),R3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Ya),R3)) ).

% Complex_mult_complex_of_real
tff(fact_9080_complex__of__real__mult__Complex,axiom,
    ! [R3: real,Xc: real,Ya: real] : aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(R3)),complex2(Xc,Ya)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),Ya)) ).

% complex_of_real_mult_Complex
tff(fact_9081_complex__of__real__add__Complex,axiom,
    ! [R3: real,Xc: real,Ya: real] : aa_complex_complex(plus_plus_complex(real_V4546457046886955230omplex(R3)),complex2(Xc,Ya)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R3),Xc),Ya) ).

% complex_of_real_add_Complex
tff(fact_9082_Complex__add__complex__of__real,axiom,
    ! [Xc: real,Ya: real,R3: real] : aa_complex_complex(plus_plus_complex(complex2(Xc,Ya)),real_V4546457046886955230omplex(R3)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),R3),Ya) ).

% Complex_add_complex_of_real
tff(fact_9083_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),zero_zero_real)
            & ( aa_real_real(cos_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ix,Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_9084_Maclaurin__cos__expansion2,axiom,
    ! [Xc: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xc)
            & ( aa_real_real(cos_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ix,Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_9085_Maclaurin__cos__expansion,axiom,
    ! [Xc: real,Nb: nat] :
    ? [T5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xc))
      & ( aa_real_real(cos_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ix,Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_9086_cos__arcsin,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xc)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_9087_arcsin__0,axiom,
    aa_real_real(arcsin,zero_zero_real) = zero_zero_real ).

% arcsin_0
tff(fact_9088_sin__arcsin,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Ya)) = Ya ) ) ) ).

% sin_arcsin
tff(fact_9089_arcsin__1,axiom,
    aa_real_real(arcsin,one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arcsin_1
tff(fact_9090_arcsin__minus__1,axiom,
    aa_real_real(arcsin,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% arcsin_minus_1
tff(fact_9091_arcsin__le__arcsin,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xc)),aa_real_real(arcsin,Ya)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_9092_arcsin__minus,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,Xc)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,Xc)) ) ) ) ).

% arcsin_minus
tff(fact_9093_arcsin__eq__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( ( aa_real_real(arcsin,Xc) = aa_real_real(arcsin,Ya) )
        <=> ( Xc = Ya ) ) ) ) ).

% arcsin_eq_iff
tff(fact_9094_arcsin__le__mono,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xc)),aa_real_real(arcsin,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ) ) ).

% arcsin_le_mono
tff(fact_9095_arcsin__less__arcsin,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xc)),aa_real_real(arcsin,Ya)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_9096_arcsin__less__mono,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xc)),aa_real_real(arcsin,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ) ) ).

% arcsin_less_mono
tff(fact_9097_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% square_fact_le_2_fact
tff(fact_9098_cos__arcsin__nonzero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xc)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_9099_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun_real_real,J2: fun_nat_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ? [B5: real] : aa_real_real(F2,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_iy(H),J2),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B5),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_9100_arcsin__lt__bounded,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Ya))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_9101_arcsin__lbound,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Ya)) ) ) ).

% arcsin_lbound
tff(fact_9102_arcsin__ubound,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arcsin_ubound
tff(fact_9103_arcsin__bounded,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Ya))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arcsin_bounded
tff(fact_9104_arcsin__sin,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xc)) = Xc ) ) ) ).

% arcsin_sin
tff(fact_9105_arcsin,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Ya))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Ya)) = Ya ) ) ) ) ).

% arcsin
tff(fact_9106_arcsin__pi,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(arcsin,Ya))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Ya)),pi)
          & ( aa_real_real(sin_real,aa_real_real(arcsin,Ya)) = Ya ) ) ) ) ).

% arcsin_pi
tff(fact_9107_arcsin__le__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xc)),Ya)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(sin_real,Ya)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_9108_le__arcsin__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Ya)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(arcsin,Xc))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Ya)),Xc) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_9109_cos__coeff__def,axiom,
    ! [X4: nat] :
      aa_nat_real(cos_coeff,X4) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X4)),zero_zero_real) ).

% cos_coeff_def
tff(fact_9110_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xc)
            & ( aa_real_real(sin_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iz,Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_9111_Maclaurin__sin__expansion4,axiom,
    ! [Xc: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ? [T5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),Xc)
          & ( aa_real_real(sin_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iz,Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_9112_Maclaurin__sin__expansion2,axiom,
    ! [Xc: real,Nb: nat] :
    ? [T5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xc))
      & ( aa_real_real(sin_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iz,Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_9113_Maclaurin__sin__expansion,axiom,
    ! [Xc: real,Nb: nat] :
    ? [T5: real] : aa_real_real(sin_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iz,Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T5),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_real(semiri5074537144036343181t_real,Nb))),pi)))),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_9114_sin__coeff__0,axiom,
    aa_nat_real(sin_coeff,zero_zero_nat) = zero_zero_real ).

% sin_coeff_0
tff(fact_9115_fact__mono__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono_nat
tff(fact_9116_fact__ge__self,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_self
tff(fact_9117_fact__less__mono__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_9118_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_9119_dvd__fact,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_9120_fact__diff__Suc,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,M))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,M)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_9121_fact__div__fact__le__pow,axiom,
    ! [R3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R3),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),R3)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R3)) ) ).

% fact_div_fact_le_pow
tff(fact_9122_sin__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(sin_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(cos_coeff,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% sin_coeff_Suc
tff(fact_9123_cos__coeff__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(cos_coeff,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_nat_real(sin_coeff,Nb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))) ).

% cos_coeff_Suc
tff(fact_9124_sin__coeff__def,axiom,
    ! [X4: nat] :
      aa_nat_real(sin_coeff,X4) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X4),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X4))) ).

% sin_coeff_def
tff(fact_9125_Maclaurin__exp__lt,axiom,
    ! [Xc: real,Nb: nat] :
      ( ( Xc != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xc))
            & ( aa_real_real(exp_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ja,Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_9126_sin__paired,axiom,
    ! [Xc: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_jb,Xc)),aa_real_real(sin_real,Xc)) ).

% sin_paired
tff(fact_9127_sin__arccos__abs,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
     => ( aa_real_real(sin_real,aa_real_real(arccos,Ya)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_arccos_abs
tff(fact_9128_sin__arccos,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xc)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_9129_exp__less__mono,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xc)),aa_real_real(exp_real,Ya)) ) ).

% exp_less_mono
tff(fact_9130_exp__less__cancel__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xc)),aa_real_real(exp_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ).

% exp_less_cancel_iff
tff(fact_9131_exp__le__cancel__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xc)),aa_real_real(exp_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ).

% exp_le_cancel_iff
tff(fact_9132_exp__eq__one__iff,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(exp_real,Xc) = one_one_real )
    <=> ( Xc = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_9133_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_9134_exp__less__one__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xc)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_9135_one__less__exp__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc) ) ).

% one_less_exp_iff
tff(fact_9136_exp__le__one__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xc)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_9137_one__le__exp__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc) ) ).

% one_le_exp_iff
tff(fact_9138_exp__ln,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xc)) = Xc ) ) ).

% exp_ln
tff(fact_9139_exp__ln__iff,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xc)) = Xc )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc) ) ).

% exp_ln_iff
tff(fact_9140_cos__arccos,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Ya)) = Ya ) ) ) ).

% cos_arccos
tff(fact_9141_arccos__0,axiom,
    aa_real_real(arccos,zero_zero_real) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% arccos_0
tff(fact_9142_exp__less__cancel,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xc)),aa_real_real(exp_real,Ya))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ).

% exp_less_cancel
tff(fact_9143_exp__total,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya)
     => ? [X3: real] : aa_real_real(exp_real,X3) = Ya ) ).

% exp_total
tff(fact_9144_exp__gt__zero,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,Xc)) ).

% exp_gt_zero
tff(fact_9145_not__exp__less__zero,axiom,
    ! [Xc: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xc)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_9146_exp__ge__zero,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xc)) ).

% exp_ge_zero
tff(fact_9147_not__exp__le__zero,axiom,
    ! [Xc: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xc)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_9148_exp__gt__one,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xc)) ) ).

% exp_gt_one
tff(fact_9149_exp__ge__add__one__self,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc)),aa_real_real(exp_real,Xc)) ).

% exp_ge_add_one_self
tff(fact_9150_arccos__le__arccos,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Ya)),aa_real_real(arccos,Xc)) ) ) ) ).

% arccos_le_arccos
tff(fact_9151_arccos__eq__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real) )
     => ( ( aa_real_real(arccos,Xc) = aa_real_real(arccos,Ya) )
      <=> ( Xc = Ya ) ) ) ).

% arccos_eq_iff
tff(fact_9152_arccos__le__mono,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Xc)),aa_real_real(arccos,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc) ) ) ) ).

% arccos_le_mono
tff(fact_9153_exp__ge__add__one__self__aux,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc)),aa_real_real(exp_real,Xc)) ) ).

% exp_ge_add_one_self_aux
tff(fact_9154_lemma__exp__total,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Ya)
     => ? [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Ya),one_one_real))
          & ( aa_real_real(exp_real,X3) = Ya ) ) ) ).

% lemma_exp_total
tff(fact_9155_ln__ge__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),aa_real_real(ln_ln_real,Xc))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Ya)),Xc) ) ) ).

% ln_ge_iff
tff(fact_9156_ln__x__over__x__mono,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Ya)),Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xc)),Xc)) ) ) ).

% ln_x_over_x_mono
tff(fact_9157_arccos__lbound,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Ya)) ) ) ).

% arccos_lbound
tff(fact_9158_arccos__less__arccos,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Ya)),aa_real_real(arccos,Xc)) ) ) ) ).

% arccos_less_arccos
tff(fact_9159_arccos__less__mono,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xc)),aa_real_real(arccos,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ) ) ).

% arccos_less_mono
tff(fact_9160_exp__le,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))) ).

% exp_le
tff(fact_9161_arccos__ubound,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Ya)),pi) ) ) ).

% arccos_ubound
tff(fact_9162_arccos__cos,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xc)) = Xc ) ) ) ).

% arccos_cos
tff(fact_9163_cos__arccos__abs,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Ya)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Ya)) = Ya ) ) ).

% cos_arccos_abs
tff(fact_9164_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi)
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_9165_exp__half__le2,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% exp_half_le2
tff(fact_9166_arccos__lt__bounded,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arccos,Ya))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Ya)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_9167_arccos__bounded,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Ya))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Ya)),pi) ) ) ) ).

% arccos_bounded
tff(fact_9168_sin__arccos__nonzero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xc)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_9169_arccos__cos2,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Xc)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xc)) = aa_real_real(uminus_uminus_real,Xc) ) ) ) ).

% arccos_cos2
tff(fact_9170_arccos__minus,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xc)) ) ) ) ).

% arccos_minus
tff(fact_9171_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_jc),one_one_real) ).

% power_half_series
tff(fact_9172_arccos,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arccos,Ya))
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Ya)),pi)
          & ( aa_real_real(cos_real,aa_real_real(arccos,Ya)) = Ya ) ) ) ) ).

% arccos
tff(fact_9173_arccos__minus__abs,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xc)) ) ) ).

% arccos_minus_abs
tff(fact_9174_sums__if_H,axiom,
    ! [G: fun_nat_real,Xc: real] :
      ( aa_real_o(sums_real(G),Xc)
     => aa_real_o(sums_real(aTP_Lamm_jd(G)),Xc) ) ).

% sums_if'
tff(fact_9175_sums__if,axiom,
    ! [G: fun_nat_real,Xc: real,F2: fun_nat_real,Ya: real] :
      ( aa_real_o(sums_real(G),Xc)
     => ( aa_real_o(sums_real(F2),Ya)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_je(G),F2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),Ya)) ) ) ).

% sums_if
tff(fact_9176_exp__bound,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_9177_real__exp__bound__lemma,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xc)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xc))) ) ) ).

% real_exp_bound_lemma
tff(fact_9178_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Xc)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,Xc)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_9179_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xc: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xc))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_9180_arccos__le__pi2,axiom,
    ! [Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Ya)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arccos_le_pi2
tff(fact_9181_Maclaurin__exp__le,axiom,
    ! [Xc: real,Nb: nat] :
    ? [T5: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xc))
      & ( aa_real_real(exp_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ja,Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_9182_exp__lower__Taylor__quadratic,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xc)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,Xc)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_9183_log__base__10__eq2,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,Xc)) ) ) ).

% log_base_10_eq2
tff(fact_9184_tanh__real__altdef,axiom,
    ! [Xc: real] : aa_real_real(tanh_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xc)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xc)))) ).

% tanh_real_altdef
tff(fact_9185_cos__paired,axiom,
    ! [Xc: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_jf,Xc)),aa_real_real(cos_real,Xc)) ).

% cos_paired
tff(fact_9186_log__base__10__eq1,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,Xc)) ) ) ).

% log_base_10_eq1
tff(fact_9187_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(aa_rea7340605660300375871l_real(minus_minus_real,Theta),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,K2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_9188_vebt__assn__raw_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Xaa: vEBT_VEBTi,Ya: assn] :
      ( ( aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Xc),Xaa) = Ya )
     => ( accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(Xc,Xaa))
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xaa = aa_o_VEBT_VEBTi(vEBT_Leafi((Ai)),(Bi)) )
                 => ( ( Ya = aa_o_assn(pure_assn,
                          ( ( (Ai) = (A3) )
                          & ( (Bi) = (B3) ) )) )
                   => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)),aa_o_VEBT_VEBTi(vEBT_Leafi((Ai)),(Bi)))) ) ) )
         => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary) )
               => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                    ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi) )
                   => ( ( Ya = aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_o_assn(pure_assn,
                                    ( ( Mmoi = Mmo )
                                    & ( Degi = Deg2 ) ))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Summary),Summaryi))),ex_ass463751140784270563_VEBTi(aa_arr7909941341466502432i_assn(aTP_Lamm_eq(Tree_list),Tree_array))) )
                     => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(Mmo,Deg2,Tree_list),Summary),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(Mmoi,Degi,Tree_array),Summaryi))) ) ) )
           => ( ! [V2: option4927543243414619207at_nat,Va2: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xaa = aa_o_VEBT_VEBTi(vEBT_Leafi((Vd3)),(Ve3)) )
                     => ( ( Ya = bot_bot_assn )
                       => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_VEB5832953974749077948T_VEBT(vEBT_Node(V2,Va2,Vb3),Vc3),aa_o_VEBT_VEBTi(vEBT_Leafi((Vd3)),(Ve3)))) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Vd3)),(Ve3)) )
                   => ! [V2: option4927543243414619207at_nat,Va2: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xaa = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3) )
                       => ( ( Ya = bot_bot_assn )
                         => ~ accp_P7675410724331315407_VEBTi(vEBT_v8524038756793281170aw_rel,produc6084888613844515218_VEBTi(aa_o_VEBT_VEBT(vEBT_Leaf((Vd3)),(Ve3)),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(V2,Va2,Vb3),Vc3))) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
tff(fact_9189_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_o_VEBT_VEBT(vEBT_Leaf((A2)),(B2))) = zero_zero_nat ).

% VEBT_internal.height.simps(1)
tff(fact_9190_merge__true__star,axiom,
    aa_assn_assn(times_times_assn(top_top_assn),top_top_assn) = top_top_assn ).

% merge_true_star
tff(fact_9191_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_9192_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
tff(fact_9193_merge__true__star__ctx,axiom,
    ! [P: assn] : aa_assn_assn(times_times_assn(top_top_assn),aa_assn_assn(times_times_assn(top_top_assn),P)) = aa_assn_assn(times_times_assn(top_top_assn),P) ).

% merge_true_star_ctx
tff(fact_9194_ent__true,axiom,
    ! [P: assn] : entails(P,top_top_assn) ).

% ent_true
tff(fact_9195_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( entails(P,Q)
     => entails(P,aa_assn_assn(times_times_assn(Q),top_top_assn)) ) ).

% ent_true_drop(2)
tff(fact_9196_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( entails(P,aa_assn_assn(times_times_assn(Q),top_top_assn))
     => entails(aa_assn_assn(times_times_assn(P),R),aa_assn_assn(times_times_assn(Q),top_top_assn)) ) ).

% ent_true_drop(1)
tff(fact_9197_ent__refl__true,axiom,
    ! [A: assn] : entails(A,aa_assn_assn(times_times_assn(A),top_top_assn)) ).

% ent_refl_true
tff(fact_9198_ent__star__mono__true,axiom,
    ! [A: assn,A5: assn,B: assn,B9: assn] :
      ( entails(A,aa_assn_assn(times_times_assn(A5),top_top_assn))
     => ( entails(B,aa_assn_assn(times_times_assn(B9),top_top_assn))
       => entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A),B)),top_top_assn),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(A5),B9)),top_top_assn)) ) ) ).

% ent_star_mono_true
tff(fact_9199_mod__star__trueI,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(P),H)
     => aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(P),top_top_assn)),H) ) ).

% mod_star_trueI
tff(fact_9200_mod__star__trueE,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(rep_assn(aa_assn_assn(times_times_assn(P),top_top_assn)),H)
     => ~ ! [H5: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(rep_assn(P),H5) ) ).

% mod_star_trueE
tff(fact_9201_mod__h__bot__iff_I2_J,axiom,
    ! [H: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(rep_assn(top_top_assn),produc7507926704131184380et_nat(H,bot_bot_set_nat)) ).

% mod_h_bot_iff(2)
tff(fact_9202_VEBT__internal_Oheight_Ocases,axiom,
    ! [Xc: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] : Xc != aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
     => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] : Xc != aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary) ) ).

% VEBT_internal.height.cases
tff(fact_9203_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_9204_Maclaurin__sin__bound,axiom,
    ! [Xc: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,Xc)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_iz,Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xc)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_9205_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A2)),aa_num_real(numeral_numeral_real,B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A2)),aa_num_int(numeral_numeral_int,B2)) ).

% floor_divide_eq_div_numeral
tff(fact_9206_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B2)) ).

% floor_one_divide_eq_div_numeral
tff(fact_9207_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_num_real(numeral_numeral_real,A2)),aa_num_real(numeral_numeral_real,B2)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A2))),aa_num_int(numeral_numeral_int,B2)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_9208_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,B2)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B2)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_9209_real__sqrt__inverse,axiom,
    ! [Xc: real] : aa_real_real(sqrt,aa_real_real(inverse_inverse_real,Xc)) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xc)) ).

% real_sqrt_inverse
tff(fact_9210_divide__real__def,axiom,
    ! [Xc: real,Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),aa_real_real(inverse_inverse_real,Ya)) ).

% divide_real_def
tff(fact_9211_inverse__powr,axiom,
    ! [Ya: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(inverse_inverse_real,Ya)),A2) = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Ya),A2)) ) ) ).

% inverse_powr
tff(fact_9212_nat__floor__neg,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xc)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_9213_floor__eq3,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xc)) = Nb ) ) ) ).

% floor_eq3
tff(fact_9214_le__nat__floor,axiom,
    ! [Xc: nat,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xc)),A2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xc),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A2))) ) ).

% le_nat_floor
tff(fact_9215_real__of__int__floor__add__one__gt,axiom,
    ! [R3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R3))),one_one_real)) ).

% real_of_int_floor_add_one_gt
tff(fact_9216_floor__eq,axiom,
    ! [Nb: int,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xc) = Nb ) ) ) ).

% floor_eq
tff(fact_9217_real__of__int__floor__add__one__ge,axiom,
    ! [R3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R3),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R3))),one_one_real)) ).

% real_of_int_floor_add_one_ge
tff(fact_9218_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D5: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D5),E2)
         => ( aa_real_o(P,D5)
           => aa_real_o(P,E2) ) )
     => ( ! [N: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_9219_real__arch__inverse,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
    <=> ? [N6: nat] :
          ( ( N6 != zero_zero_nat )
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N6)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N6))),E) ) ) ).

% real_arch_inverse
tff(fact_9220_forall__pos__mono,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D5: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D5),E2)
         => ( aa_real_o(P,D5)
           => aa_real_o(P,E2) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono
tff(fact_9221_real__of__int__floor__gt__diff__one,axiom,
    ! [R3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R3),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R3))) ).

% real_of_int_floor_gt_diff_one
tff(fact_9222_real__of__int__floor__ge__diff__one,axiom,
    ! [R3: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R3),one_one_real)),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,R3))) ).

% real_of_int_floor_ge_diff_one
tff(fact_9223_sqrt__divide__self__eq,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xc)),Xc) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xc)) ) ) ).

% sqrt_divide_self_eq
tff(fact_9224_ln__inverse,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xc)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,Xc)) ) ) ).

% ln_inverse
tff(fact_9225_floor__eq4,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xc)) = Nb ) ) ) ).

% floor_eq4
tff(fact_9226_floor__eq2,axiom,
    ! [Nb: int,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Nb)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xc) = Nb ) ) ) ).

% floor_eq2
tff(fact_9227_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_int_real(ring_1_of_int_real,B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A2)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_9228_log__inverse,axiom,
    ! [A2: real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
     => ( ( A2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A2),aa_real_real(inverse_inverse_real,Xc)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A2),Xc)) ) ) ) ) ).

% log_inverse
tff(fact_9229_exp__plus__inverse__exp,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xc)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xc)))) ).

% exp_plus_inverse_exp
tff(fact_9230_plus__inverse__ge__2,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(inverse_inverse_real,Xc))) ) ).

% plus_inverse_ge_2
tff(fact_9231_real__inv__sqrt__pow2,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xc))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xc) ) ) ).

% real_inv_sqrt_pow2
tff(fact_9232_tan__cot,axiom,
    ! [Xc: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xc)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,Xc)) ).

% tan_cot
tff(fact_9233_real__le__x__sinh,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xc)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xc)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_9234_real__le__abs__sinh,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xc)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xc)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_9235_floor__log__eq__powr__iff,axiom,
    ! [Xc: real,B2: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_int_real(ring_1_of_int_real,K))),Xc)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(K),one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_9236_powr__real__of__int,axiom,
    ! [Xc: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),aa_int_real(ring_1_of_int_real,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_int_nat(nat2,Nb)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Nb))))) ) ) ).

% powr_real_of_int
tff(fact_9237_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_int_int(plus_plus_int(aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_9238_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_9239_divide__complex__def,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_complex(divide1717551699836669952omplex(Xc),Ya) = aa_complex_complex(times_times_complex(Xc),aa_complex_complex(invers8013647133539491842omplex,Ya)) ).

% divide_complex_def
tff(fact_9240_real__scaleR__def,axiom,
    ! [A2: real,Xc: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A2),Xc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),Xc) ).

% real_scaleR_def
tff(fact_9241_complex__scaleR,axiom,
    ! [R3: real,A2: real,B2: real] : aa_complex_complex(real_V2046097035970521341omplex(R3),complex2(A2,B2)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),A2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),B2)) ).

% complex_scaleR
tff(fact_9242_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_9243_prod__Suc__fact,axiom,
    ! [Nb: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_9244_fact__eq__fact__times,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),groups708209901874060359at_nat(aTP_Lamm_cn,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb)),M))) ) ) ).

% fact_eq_fact_times
tff(fact_9245_fact__div__fact,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),M)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = groups708209901874060359at_nat(aTP_Lamm_cn,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),M)) ) ) ).

% fact_div_fact
tff(fact_9246_complex__inverse,axiom,
    ! [A2: real,B2: real] : aa_complex_complex(invers8013647133539491842omplex,complex2(A2,B2)) = complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_9247_sinh__real__zero__iff,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(sinh_real,Xc) = zero_zero_real )
    <=> ( Xc = zero_zero_real ) ) ).

% sinh_real_zero_iff
tff(fact_9248_sinh__real__less__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xc)),aa_real_real(sinh_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ).

% sinh_real_less_iff
tff(fact_9249_sinh__real__le__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xc)),aa_real_real(sinh_real,Ya))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ).

% sinh_real_le_iff
tff(fact_9250_sinh__real__pos__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc) ) ).

% sinh_real_pos_iff
tff(fact_9251_sinh__real__neg__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xc)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_9252_sinh__real__nonpos__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xc)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_9253_sinh__real__nonneg__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc) ) ).

% sinh_real_nonneg_iff
tff(fact_9254_sinh__less__cosh__real,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xc)),aa_real_real(cosh_real,Xc)) ).

% sinh_less_cosh_real
tff(fact_9255_sinh__le__cosh__real,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xc)),aa_real_real(cosh_real,Xc)) ).

% sinh_le_cosh_real
tff(fact_9256_cosh__real__nonzero,axiom,
    ! [Xc: real] : aa_real_real(cosh_real,Xc) != zero_zero_real ).

% cosh_real_nonzero
tff(fact_9257_cosh__real__pos,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,Xc)) ).

% cosh_real_pos
tff(fact_9258_cosh__real__nonpos__le__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xc)),aa_real_real(cosh_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),Xc) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_9259_cosh__real__nonneg__le__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xc)),aa_real_real(cosh_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_9260_cosh__real__nonneg,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xc)) ).

% cosh_real_nonneg
tff(fact_9261_cosh__real__ge__1,axiom,
    ! [Xc: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,Xc)) ).

% cosh_real_ge_1
tff(fact_9262_cosh__real__nonpos__less__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ya),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xc)),aa_real_real(cosh_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Ya),Xc) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_9263_cosh__real__nonneg__less__iff,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xc)),aa_real_real(cosh_real,Ya))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_9264_cosh__real__strict__mono,axiom,
    ! [Xc: real,Ya: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xc)),aa_real_real(cosh_real,Ya)) ) ) ).

% cosh_real_strict_mono
tff(fact_9265_prod__int__eq,axiom,
    ! [I: nat,J2: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J2)) = groups1705073143266064639nt_int(aTP_Lamm_co,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J2))) ).

% prod_int_eq
tff(fact_9266_arcosh__cosh__real,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xc)) = Xc ) ) ).

% arcosh_cosh_real
tff(fact_9267_prod__int__plus__eq,axiom,
    ! [I: nat,J2: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2))) = groups1705073143266064639nt_int(aTP_Lamm_co,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J2)))) ).

% prod_int_plus_eq
tff(fact_9268_cosh__ln__real,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(inverse_inverse_real,Xc))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_9269_sinh__ln__real,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),aa_real_real(inverse_inverse_real,Xc))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_9270_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K),
        zero_zero_nat,
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),one_one_nat)),Nb),one_one_nat)),aa_nat_nat(semiri1408675320244567234ct_nat,K))) ) ).

% binomial_code
tff(fact_9271_binomial__Suc__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Nb) = aa_nat_nat(suc,Nb) ).

% binomial_Suc_n
tff(fact_9272_finite__atMost,axiom,
    ! [K: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% finite_atMost
tff(fact_9273_binomial__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,zero_zero_nat)) = Nb ).

% binomial_1
tff(fact_9274_binomial__0__Suc,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% binomial_0_Suc
tff(fact_9275_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_9276_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ).

% binomial_Suc_Suc
tff(fact_9277_binomial__n__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),zero_zero_nat) = one_one_nat ).

% binomial_n_0
tff(fact_9278_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_9279_atMost__0,axiom,
    aa_nat_set_nat(set_ord_atMost_nat,zero_zero_nat) = insert_nat(zero_zero_nat,bot_bot_set_nat) ).

% atMost_0
tff(fact_9280_sum__choose__lower,axiom,
    ! [R3: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jg,R3)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,R3),Nb))),Nb) ).

% sum_choose_lower
tff(fact_9281_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jh,Nb)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)),one_one_nat)),M) ).

% choose_rising_sum(2)
tff(fact_9282_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jh,Nb)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),M)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_9283_sum__choose__upper,axiom,
    ! [M: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ji,M)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,M)) ).

% sum_choose_upper
tff(fact_9284_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = zero_zero_nat ) ) ).

% binomial_eq_0
tff(fact_9285_Suc__times__binomial,axiom,
    ! [K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ).

% Suc_times_binomial
tff(fact_9286_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K))),aa_nat_nat(suc,K)) ).

% Suc_times_binomial_eq
tff(fact_9287_binomial__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_9288_choose__mult__lemma,axiom,
    ! [M: nat,R3: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R3)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),K)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R3)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),R3)),M)) ).

% choose_mult_lemma
tff(fact_9289_binomial__le__pow,axiom,
    ! [R3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R3),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),R3)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),R3)) ) ).

% binomial_le_pow
tff(fact_9290_atMost__atLeast0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,Nb) = aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb) ).

% atMost_atLeast0
tff(fact_9291_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_ord_atMost_nat,K) ).

% lessThan_Suc_atMost
tff(fact_9292_sum__choose__diagonal,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jj(M),Nb)),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),M) ) ) ).

% sum_choose_diagonal
tff(fact_9293_vandermonde,axiom,
    ! [M: nat,Nb: nat,R3: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jk(M),Nb),R3)),aa_nat_set_nat(set_ord_atMost_nat,R3)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb)),R3) ).

% vandermonde
tff(fact_9294_atMost__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K)) = insert_nat(aa_nat_nat(suc,K),aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% atMost_Suc
tff(fact_9295_choose__row__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% choose_row_sum
tff(fact_9296_binomial,axiom,
    ! [A2: nat,B2: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jl(A2),B2),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_9297_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ).

% zero_less_binomial
tff(fact_9298_Suc__times__binomial__add,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,A2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))),aa_nat_nat(suc,A2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B2)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B2))),A2)) ).

% Suc_times_binomial_add
tff(fact_9299_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))),aa_nat_nat(suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_9300_choose__mult,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),M)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),M)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_9301_binomial__absorb__comp,axiom,
    ! [Nb: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).

% binomial_absorb_comp
tff(fact_9302_choose__square__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jm,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb) ).

% choose_square_sum
tff(fact_9303_atMost__nat__numeral,axiom,
    ! [K: num] : aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(numeral_numeral_nat,K)) = insert_nat(aa_num_nat(numeral_numeral_nat,K),aa_nat_set_nat(set_ord_atMost_nat,aa_num_nat(pred_numeral,K))) ).

% atMost_nat_numeral
tff(fact_9304_binomial__absorption,axiom,
    ! [K: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ).

% binomial_absorption
tff(fact_9305_binomial__r__part__sum,axiom,
    ! [M: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,M)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)) ).

% binomial_r_part_sum
tff(fact_9306_choose__linear__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jn,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ).

% choose_linear_sum
tff(fact_9307_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_9308_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% binomial_maximum
tff(fact_9309_binomial__antimono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_9310_binomial__mono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K4)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K4)) ) ) ).

% binomial_mono
tff(fact_9311_binomial__maximum_H,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_9312_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% binomial_le_pow2
tff(fact_9313_choose__reduce__nat,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_9314_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% times_binomial_minus1_eq
tff(fact_9315_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_9316_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_9317_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K4)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K4)) ) ) ).

% binomial_strict_mono
tff(fact_9318_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K4)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K4)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_9319_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% central_binomial_odd
tff(fact_9320_binomial__addition__formula,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_nat_nat(suc,K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_9321_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,zero_zero_nat)),Nb) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atMost_nat,Nb),insert_nat(zero_zero_nat,bot_bot_set_nat)) ).

% atLeast1_atMost_eq_remove0
tff(fact_9322_choose__two,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% choose_two
tff(fact_9323_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun_nat_nat,Nb: nat,B2: fun_nat_nat,Xc: nat] :
      ( ! [I5: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),I5)
         => ( aa_nat_nat(A2,I5) = zero_zero_nat ) )
     => ( ! [J3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J3)
           => ( aa_nat_nat(B2,J3) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jo(A2),Xc)),aa_nat_set_nat(set_ord_atMost_nat,M))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_jo(B2),Xc)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_jq(A2),B2),Xc)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_9324_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb)))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_9325_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_9326_exp__two__pi__i,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),real_V4546457046886955230omplex(pi))),imaginary_unit)) = one_one_complex ).

% exp_two_pi_i
tff(fact_9327_exp__two__pi__i_H,axiom,
    aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) = one_one_complex ).

% exp_two_pi_i'
tff(fact_9328_divide__i,axiom,
    ! [Xc: complex] : aa_complex_complex(divide1717551699836669952omplex(Xc),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),Xc) ).

% divide_i
tff(fact_9329_divide__numeral__i,axiom,
    ! [Z: complex,Nb: num] : aa_complex_complex(divide1717551699836669952omplex(Z),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Nb)),imaginary_unit)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Z))),numera6690914467698888265omplex(Nb)) ).

% divide_numeral_i
tff(fact_9330_power2__i,axiom,
    aa_nat_complex(power_power_complex(imaginary_unit),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% power2_i
tff(fact_9331_i__even__power,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(imaginary_unit),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb) ).

% i_even_power
tff(fact_9332_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero_real,one_one_real) ).

% imaginary_unit.code
tff(fact_9333_Complex__eq__i,axiom,
    ! [Xc: real,Ya: real] :
      ( ( complex2(Xc,Ya) = imaginary_unit )
    <=> ( ( Xc = zero_zero_real )
        & ( Ya = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_9334_i__complex__of__real,axiom,
    ! [R3: real] : aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(R3)) = complex2(zero_zero_real,R3) ).

% i_complex_of_real
tff(fact_9335_complex__of__real__i,axiom,
    ! [R3: real] : aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(R3)),imaginary_unit) = complex2(zero_zero_real,R3) ).

% complex_of_real_i
tff(fact_9336_csqrt__ii,axiom,
    aa_complex_complex(csqrt,imaginary_unit) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(one_one_complex),imaginary_unit)),real_V4546457046886955230omplex(aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt_ii
tff(fact_9337_Arg__minus__ii,axiom,
    aa_complex_real(arg,aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_minus_ii
tff(fact_9338_Arg__ii,axiom,
    aa_complex_real(arg,imaginary_unit) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% Arg_ii
tff(fact_9339_power2__csqrt,axiom,
    ! [Z: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(csqrt,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z ).

% power2_csqrt
tff(fact_9340_Arg__zero,axiom,
    aa_complex_real(arg,zero_zero_complex) = zero_zero_real ).

% Arg_zero
tff(fact_9341_of__real__sqrt,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( real_V4546457046886955230omplex(aa_real_real(sqrt,Xc)) = aa_complex_complex(csqrt,real_V4546457046886955230omplex(Xc)) ) ) ).

% of_real_sqrt
tff(fact_9342_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ).

% Arg_bounded
tff(fact_9343_cis__minus__pi__half,axiom,
    cis(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit) ).

% cis_minus_pi_half
tff(fact_9344_cot__less__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xc)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_9345_cis__zero,axiom,
    cis(zero_zero_real) = one_one_complex ).

% cis_zero
tff(fact_9346_cot__pi,axiom,
    aa_real_real(cot_real,pi) = zero_zero_real ).

% cot_pi
tff(fact_9347_cot__npi,axiom,
    ! [Nb: nat] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi)) = zero_zero_real ).

% cot_npi
tff(fact_9348_cis__pi__half,axiom,
    cis(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = imaginary_unit ).

% cis_pi_half
tff(fact_9349_cis__2pi,axiom,
    cis(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)) = one_one_complex ).

% cis_2pi
tff(fact_9350_cot__periodic,axiom,
    ! [Xc: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,Xc) ).

% cot_periodic
tff(fact_9351_cis__mult,axiom,
    ! [A2: real,B2: real] : aa_complex_complex(times_times_complex(cis(A2)),cis(B2)) = cis(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)) ).

% cis_mult
tff(fact_9352_cis__divide,axiom,
    ! [A2: real,B2: real] : aa_complex_complex(divide1717551699836669952omplex(cis(A2)),cis(B2)) = cis(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),B2)) ).

% cis_divide
tff(fact_9353_DeMoivre,axiom,
    ! [A2: real,Nb: nat] : aa_nat_complex(power_power_complex(cis(A2)),Nb) = cis(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A2)) ).

% DeMoivre
tff(fact_9354_cot__gt__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,Xc)) ) ) ).

% cot_gt_zero
tff(fact_9355_tan__cot_H,axiom,
    ! [Xc: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xc)) = aa_real_real(cot_real,Xc) ).

% tan_cot'
tff(fact_9356_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_complex_o(bij_betw_nat_complex(aTP_Lamm_jr(Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb)),collect_complex(aTP_Lamm_fr(Nb))) ) ).

% bij_betw_roots_unity
tff(fact_9357_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique5052692396658037445od_int(aa_num_num(bitM,M),aa_num_num(bit0,one)) = product_Pair_int_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_9358_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(bit0,K)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,K)) ).

% pred_numeral_simps(2)
tff(fact_9359_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_9360_semiring__norm_I28_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(bit0,Nb)) ).

% semiring_norm(28)
tff(fact_9361_semiring__norm_I27_J,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(bitM,Nb)) ).

% semiring_norm(27)
tff(fact_9362_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Nb)) = aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,aa_num_num(bitM,Nb))) ).

% eval_nat_numeral(2)
tff(fact_9363_BitM__plus__one,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,Nb)),one) = aa_num_num(bit0,Nb) ).

% BitM_plus_one
tff(fact_9364_one__plus__BitM,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% one_plus_BitM
tff(fact_9365_set__decode__0,axiom,
    ! [Xc: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),aa_nat_set_nat(nat_set_decode,Xc))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xc) ) ).

% set_decode_0
tff(fact_9366_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_9367_set__decode__zero,axiom,
    aa_nat_set_nat(nat_set_decode,zero_zero_nat) = bot_bot_set_nat ).

% set_decode_zero
tff(fact_9368_set__decode__Suc,axiom,
    ! [Nb: nat,Xc: nat] :
      ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Nb)),aa_nat_set_nat(nat_set_decode,Xc))
    <=> aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_9369_subset__decode__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,M)),aa_nat_set_nat(nat_set_decode,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb) ) ).

% subset_decode_imp_le
tff(fact_9370_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ aa_set_nat_o(member_nat(Nb),aa_nat_set_nat(nat_set_decode,Z))
     => ( aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Z)) = insert_nat(Nb,aa_nat_set_nat(nat_set_decode,Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_9371_set__decode__def,axiom,
    ! [Xc: nat] : aa_nat_set_nat(nat_set_decode,Xc) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_js,Xc)) ).

% set_decode_def
tff(fact_9372_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_set_complex_o(bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(aa_real_real(aa_nat_fun_real_real(root,Nb),aa_complex_real(real_V1022390504157884413omplex,C2)))),cis(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C2)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aTP_Lamm_fr(Nb))),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_jt(C2),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_9373_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
        ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
        aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_9374_and__int_Oelims,axiom,
    ! [Xc: int,Xaa: int,Ya: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xc),Xaa) = Ya )
     => ( Ya = $ite(
            ( aa_set_int_o(member_int(Xc),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xc)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
            aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xc)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xc),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_9375_real__root__zero,axiom,
    ! [Nb: nat] : aa_real_real(aa_nat_fun_real_real(root,Nb),zero_zero_real) = zero_zero_real ).

% real_root_zero
tff(fact_9376_real__root__Suc__0,axiom,
    ! [Xc: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xc) = Xc ).

% real_root_Suc_0
tff(fact_9377_real__root__eq__iff,axiom,
    ! [Nb: nat,Xc: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xc) = aa_real_real(aa_nat_fun_real_real(root,Nb),Ya) )
      <=> ( Xc = Ya ) ) ) ).

% real_root_eq_iff
tff(fact_9378_root__0,axiom,
    ! [Xc: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xc) = zero_zero_real ).

% root_0
tff(fact_9379_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_9380_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% and_negative_int_iff
tff(fact_9381_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xc) = zero_zero_real )
      <=> ( Xc = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_9382_real__root__less__iff,axiom,
    ! [Nb: nat,Xc: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_real_real(aa_nat_fun_real_real(root,Nb),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya) ) ) ).

% real_root_less_iff
tff(fact_9383_real__root__le__iff,axiom,
    ! [Nb: nat,Xc: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_real_real(aa_nat_fun_real_real(root,Nb),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya) ) ) ).

% real_root_le_iff
tff(fact_9384_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),one_one_real) = one_one_real ) ) ).

% real_root_one
tff(fact_9385_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xc) = one_one_real )
      <=> ( Xc = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_9386_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Ya) ) ) ).

% real_root_gt_0_iff
tff(fact_9387_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9388_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya) ) ) ).

% real_root_ge_0_iff
tff(fact_9389_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9390_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Ya) ) ) ).

% real_root_gt_1_iff
tff(fact_9391_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9392_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Ya) ) ) ).

% real_root_ge_1_iff
tff(fact_9393_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9394_and__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_9395_and__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_9396_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),Nb) = Xc ) ) ) ).

% real_root_pow_pos2
tff(fact_9397_and__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_9398_and__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_9399_real__root__inverse,axiom,
    ! [Nb: nat,Xc: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,Xc)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)) ).

% real_root_inverse
tff(fact_9400_real__root__divide,axiom,
    ! [Nb: nat,Xc: real,Ya: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_real_real(aa_nat_fun_real_real(root,Nb),Ya)) ).

% real_root_divide
tff(fact_9401_real__root__mult__exp,axiom,
    ! [M: nat,Nb: nat,Xc: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)),Xc) = aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)) ).

% real_root_mult_exp
tff(fact_9402_real__root__mult,axiom,
    ! [Nb: nat,Xc: real,Ya: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_real_real(aa_nat_fun_real_real(root,Nb),Ya)) ).

% real_root_mult
tff(fact_9403_real__root__commute,axiom,
    ! [M: nat,Nb: nat,Xc: real] : aa_real_real(aa_nat_fun_real_real(root,M),aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)) = aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_nat_fun_real_real(root,M),Xc)) ).

% real_root_commute
tff(fact_9404_real__root__minus,axiom,
    ! [Nb: nat,Xc: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(uminus_uminus_real,Xc)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)) ).

% real_root_minus
tff(fact_9405_real__root__pos__pos__le,axiom,
    ! [Xc: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)) ) ).

% real_root_pos_pos_le
tff(fact_9406_AND__lower,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(Xc),Ya)) ) ).

% AND_lower
tff(fact_9407_AND__upper1,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Xc),Ya)),Xc) ) ).

% AND_upper1
tff(fact_9408_AND__upper2,axiom,
    ! [Ya: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Xc),Ya)),Ya) ) ).

% AND_upper2
tff(fact_9409_AND__upper1_H,axiom,
    ! [Ya: int,Z: int,Yaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Ya),Yaa)),Z) ) ) ).

% AND_upper1'
tff(fact_9410_AND__upper2_H,axiom,
    ! [Ya: int,Z: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ya),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Xc),Ya)),Z) ) ) ).

% AND_upper2'
tff(fact_9411_real__root__less__mono,axiom,
    ! [Nb: nat,Xc: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_real_real(aa_nat_fun_real_real(root,Nb),Ya)) ) ) ).

% real_root_less_mono
tff(fact_9412_real__root__le__mono,axiom,
    ! [Nb: nat,Xc: real,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),Ya)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_real_real(aa_nat_fun_real_real(root,Nb),Ya)) ) ) ).

% real_root_le_mono
tff(fact_9413_real__root__power,axiom,
    ! [Nb: nat,Xc: real,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),K) ) ) ).

% real_root_power
tff(fact_9414_real__root__abs,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(abs_abs_real,Xc)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)) ) ) ).

% real_root_abs
tff(fact_9415_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(K),L)),K) ) ).

% and_less_eq
tff(fact_9416_AND__upper1_H_H,axiom,
    ! [Ya: int,Z: int,Yaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Ya),Yaa)),Z) ) ) ).

% AND_upper1''
tff(fact_9417_AND__upper2_H_H,axiom,
    ! [Ya: int,Z: int,Xc: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Xc),Ya)),Z) ) ) ).

% AND_upper2''
tff(fact_9418_real__root__gt__zero,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)) ) ) ).

% real_root_gt_zero
tff(fact_9419_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N5: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N5),Xc)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9420_sqrt__def,axiom,
    sqrt = aa_nat_fun_real_real(root,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% sqrt_def
tff(fact_9421_root__abs__power,axiom,
    ! [Nb: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),Nb))) = aa_real_real(abs_abs_real,Ya) ) ) ).

% root_abs_power
tff(fact_9422_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ) ) ).

% even_and_iff_int
tff(fact_9423_real__root__pos__pos,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)) ) ) ).

% real_root_pos_pos
tff(fact_9424_odd__real__root__pow,axiom,
    ! [Nb: nat,Xc: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),Nb) = Xc ) ) ).

% odd_real_root_pow
tff(fact_9425_odd__real__root__unique,axiom,
    ! [Nb: nat,Ya: real,Xc: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),Nb) = Xc )
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xc) = Ya ) ) ) ).

% odd_real_root_unique
tff(fact_9426_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xc: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb)) = Xc ) ) ).

% odd_real_root_power_cancel
tff(fact_9427_real__root__strict__increasing,axiom,
    ! [Nb: nat,N5: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_real_real(aa_nat_fun_real_real(root,N5),Xc)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9428_real__root__decreasing,axiom,
    ! [Nb: nat,N5: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xc)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N5),Xc)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)) ) ) ) ).

% real_root_decreasing
tff(fact_9429_real__root__pow__pos,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),Nb) = Xc ) ) ) ).

% real_root_pow_pos
tff(fact_9430_real__root__power__cancel,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb)) = Xc ) ) ) ).

% real_root_power_cancel
tff(fact_9431_real__root__pos__unique,axiom,
    ! [Nb: nat,Ya: real,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Ya)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),Nb) = Xc )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xc) = Ya ) ) ) ) ).

% real_root_pos_unique
tff(fact_9432_real__root__increasing,axiom,
    ! [Nb: nat,N5: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_real_real(aa_nat_fun_real_real(root,N5),Xc)) ) ) ) ) ).

% real_root_increasing
tff(fact_9433_log__root,axiom,
    ! [Nb: nat,A2: real,B2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_nat_fun_real_real(root,Nb),A2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),A2)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_9434_log__base__root,axiom,
    ! [Nb: nat,B2: real,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Nb),B2)),Xc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xc)) ) ) ) ).

% log_base_root
tff(fact_9435_ln__root,axiom,
    ! [Nb: nat,B2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Nb),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B2)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_9436_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_9437_root__powr__inverse,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xc) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xc),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_9438_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
        ( ( K = zero_zero_int )
        | ( L = zero_zero_int ) ),
        zero_zero_int,
        $ite(
          K = aa_int_int(uminus_uminus_int,one_one_int),
          L,
          $ite(L = aa_int_int(uminus_uminus_int,one_one_int),K,aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% and_int_unfold
tff(fact_9439_and__int_Opelims,axiom,
    ! [Xc: int,Xaa: int,Ya: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xc),Xaa) = Ya )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(Xc,Xaa))
       => ~ ( ( Ya = $ite(
                  ( aa_set_int_o(member_int(Xc),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & aa_set_int_o(member_int(Xaa),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xc)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
                  aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xc)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xc),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(Xc,Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_9440_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K,L))
     => ( aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
            ( aa_set_int_o(member_int(K),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & aa_set_int_o(member_int(L),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
            aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_9441_forall__finite_I3_J,axiom,
    ! [Xc: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,aa_nat_nat(suc,Xc)))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Xc))
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% forall_finite(3)
tff(fact_9442_and__nat__numerals_I1_J,axiom,
    ! [Ya: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ya))) = zero_zero_nat ).

% and_nat_numerals(1)
tff(fact_9443_and__nat__numerals_I3_J,axiom,
    ! [Xc: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xc))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_9444_and__nat__numerals_I2_J,axiom,
    ! [Ya: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ya))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_9445_and__nat__numerals_I4_J,axiom,
    ! [Xc: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xc))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_9446_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% and_Suc_0_eq
tff(fact_9447_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Suc_0_and_eq
tff(fact_9448_and__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Nb) = aa_int_nat(nat2,aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% and_nat_def
tff(fact_9449_forall__finite_I1_J,axiom,
    ! [P: fun_nat_o,I6: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I6),zero_zero_nat)
     => aa_nat_o(P,I6) ) ).

% forall_finite(1)
tff(fact_9450_and__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Nb) = $ite(
        ( ( M = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_9451_and__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_9452_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(A0,A1))
     => ( ! [K2: int,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K2,L3))
           => ( ( ~ ( aa_set_int_o(member_int(K2),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & aa_set_int_o(member_int(L3),insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L3),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) )
             => aa_int_o(aa_int_fun_int_o(P,K2),L3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_9453_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xc: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Xc))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Xc)
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_9454_forall__finite_I2_J,axiom,
    ! [P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,zero_zero_nat))
         => aa_nat_o(P,I2) )
    <=> aa_nat_o(P,zero_zero_nat) ) ).

% forall_finite(2)
tff(fact_9455_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(A0,A1))
     => ( ! [I5: int,J3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I5,J3))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I5),J3)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(plus_plus_int(I5),one_one_int)),J3) )
             => aa_int_o(aa_int_fun_int_o(P,I5),J3) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_9456_uint32_Osize__eq,axiom,
    ! [P3: uint32] : size_size_uint32(P3) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))) ).

% uint32.size_eq
tff(fact_9457_pred__numeral__inc,axiom,
    ! [K: num] : aa_num_nat(pred_numeral,aa_num_num(inc,K)) = aa_num_nat(numeral_numeral_nat,K) ).

% pred_numeral_inc
tff(fact_9458_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_9459_take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(times_times_int(K),L)) ).

% take_bit_mult
tff(fact_9460_take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(minus_minus_int(K),L)) ).

% take_bit_diff
tff(fact_9461_take__bit__nat__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ) ).

% take_bit_nat_eq
tff(fact_9462_nat__take__bit__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_int_nat(nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_9463_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_9464_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Nb: nat,Q3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q3)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_9465_take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,K)) ).

% take_bit_minus
tff(fact_9466_add__inc,axiom,
    ! [Xc: num,Ya: num] : aa_num_num(plus_plus_num(Xc),aa_num_num(inc,Ya)) = aa_num_num(inc,aa_num_num(plus_plus_num(Xc),Ya)) ).

% add_inc
tff(fact_9467_num__induct,axiom,
    ! [P: fun_num_o,Xc: num] :
      ( aa_num_o(P,one)
     => ( ! [X3: num] :
            ( aa_num_o(P,X3)
           => aa_num_o(P,aa_num_num(inc,X3)) )
       => aa_num_o(P,Xc) ) ) ).

% num_induct
tff(fact_9468_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(M),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_9469_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_9470_take__bit__nonnegative,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% take_bit_nonnegative
tff(fact_9471_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_int_int(bit_se2923211474154528505it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% take_bit_int_greater_self_iff
tff(fact_9472_not__take__bit__negative,axiom,
    ! [Nb: nat,K: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),zero_zero_int) ).

% not_take_bit_negative
tff(fact_9473_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_9474_inc_Osimps_I2_J,axiom,
    ! [Xc: num] : aa_num_num(inc,aa_num_num(bit0,Xc)) = aa_num_num(bit1,Xc) ).

% inc.simps(2)
tff(fact_9475_inc_Osimps_I3_J,axiom,
    ! [Xc: num] : aa_num_num(inc,aa_num_num(bit1,Xc)) = aa_num_num(bit0,aa_num_num(inc,Xc)) ).

% inc.simps(3)
tff(fact_9476_add__One,axiom,
    ! [Xc: num] : aa_num_num(plus_plus_num(Xc),one) = aa_num_num(inc,Xc) ).

% add_One
tff(fact_9477_inc__BitM__eq,axiom,
    ! [Nb: num] : aa_num_num(inc,aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% inc_BitM_eq
tff(fact_9478_BitM__inc__eq,axiom,
    ! [Nb: num] : aa_num_num(bitM,aa_num_num(inc,Nb)) = aa_num_num(bit1,Nb) ).

% BitM_inc_eq
tff(fact_9479_mult__inc,axiom,
    ! [Xc: num,Ya: num] : aa_num_num(times_times_num(Xc),aa_num_num(inc,Ya)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(Xc),Ya)),Xc) ).

% mult_inc
tff(fact_9480_take__bit__decr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) != zero_zero_int )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(minus_minus_int(K),one_one_int)) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_9481_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M) = M )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_9482_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,M: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_9483_take__bit__nat__eq__self,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_9484_take__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_def
tff(fact_9485_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_9486_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_less_exp
tff(fact_9487_take__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_int_int(modulo_modulo_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_int_def
tff(fact_9488_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_9489_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),M)),M)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_9490_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_9491_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),K)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_9492_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se2923211474154528505it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_9493_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_9494_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = K )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_9495_take__bit__incr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) != aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(plus_plus_int(K),one_one_int)) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ) ).

% take_bit_incr_eq
tff(fact_9496_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(times_times_int(aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_9497_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(minus_minus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_9498_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(plus_plus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_9499_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_ju(L),Qr) ).

% divmod_step_nat_def
tff(fact_9500_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(plus_plus_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_9501_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_jv(L),Qr) ).

% divmod_step_int_def
tff(fact_9502_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,K)) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_9503_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_jw(L),Qr) ).

% divmod_step_integer_def
tff(fact_9504_divmod__nat__if,axiom,
    ! [M: nat,Nb: nat] :
      divmod_nat(M,Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M),Nb) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),M),
        produc2626176000494625587at_nat(aTP_Lamm_jx,divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_9505_divmod__integer_H__def,axiom,
    ! [M: num,Nb: num] : unique3479559517661332726nteger(M,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Nb)),modulo364778990260209775nteger(numera6620942414471956472nteger(M),numera6620942414471956472nteger(Nb))) ).

% divmod_integer'_def
tff(fact_9506_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
tff(fact_9507_divmod__nat__def,axiom,
    ! [M: nat,Nb: nat] : divmod_nat(M,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb)) ).

% divmod_nat_def
tff(fact_9508_abs__integer__code,axiom,
    ! [K: code_integer] :
      abs_abs_Code_integer(K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),uminus1351360451143612070nteger(K),K) ).

% abs_integer_code
tff(fact_9509_less__integer__code_I1_J,axiom,
    ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),zero_z3403309356797280102nteger) ).

% less_integer_code(1)
tff(fact_9510_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,L) = uminus1351360451143612070nteger(L) ).

% minus_integer_code(2)
tff(fact_9511_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : minus_8373710615458151222nteger(K,zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_9512_integer__of__int__code,axiom,
    ! [K: int] :
      code_integer_of_int(K) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
        uminus1351360451143612070nteger(code_integer_of_int(aa_int_int(uminus_uminus_int,K))),
        $ite(
          K = zero_zero_int,
          zero_z3403309356797280102nteger,
          $let(
            l: code_integer,
            l:= times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),code_integer_of_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_9513_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_jy(D2))) ).

% int_ge_less_than2_def
tff(fact_9514_divide__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xc: int] : divide6298287555418463151nteger(code_integer_of_int(Xaa),code_integer_of_int(Xc)) = code_integer_of_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),Xc)) ).

% divide_integer.abs_eq
tff(fact_9515_less__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xc: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,code_integer_of_int(Xaa)),code_integer_of_int(Xc))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xaa),Xc) ) ).

% less_integer.abs_eq
tff(fact_9516_minus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xc: int] : minus_8373710615458151222nteger(code_integer_of_int(Xaa),code_integer_of_int(Xc)) = code_integer_of_int(aa_int_int(minus_minus_int(Xaa),Xc)) ).

% minus_integer.abs_eq
tff(fact_9517_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_jz(D2))) ).

% int_ge_less_than_def
tff(fact_9518_arctan__def,axiom,
    ! [Ya: real] : aa_real_real(arctan,Ya) = the_real(aa_real_fun_real_o(aTP_Lamm_ka,Ya)) ).

% arctan_def
tff(fact_9519_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(modulo_modulo_int(K),L) = $ite(
        L = zero_zero_int,
        K,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_int_int(minus_minus_int(aa_int_int(times_times_int(aa_int_int(abs_abs_int,L)),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_9520_arcsin__def,axiom,
    ! [Ya: real] : aa_real_real(arcsin,Ya) = the_real(aa_real_fun_real_o(aTP_Lamm_kb,Ya)) ).

% arcsin_def
tff(fact_9521_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_9522_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
       => ( aa_int_int(sgn_sgn_int,aa_int_int(modulo_modulo_int(K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_9523_ln__neg__is__const,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xc) = the_real(aTP_Lamm_ch) ) ) ).

% ln_neg_is_const
tff(fact_9524_zsgn__def,axiom,
    ! [I: int] :
      aa_int_int(sgn_sgn_int,I) = $ite(
        I = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_9525_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_9526_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_9527_arccos__def,axiom,
    ! [Ya: real] : aa_real_real(arccos,Ya) = the_real(aa_real_fun_real_o(aTP_Lamm_kc,Ya)) ).

% arccos_def
tff(fact_9528_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K: int,Q3: int] :
      ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,L))
       => ( ( K = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q3),L)),R3) )
         => aa_Pro4592068669816069595_int_o(eucl_rel_int(K,L),product_Pair_int_int(Q3,R3)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_9529_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( aa_int_int(sgn_sgn_int,K) != aa_int_int(sgn_sgn_int,L) )
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)))),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_9530_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
     => ( ( ( A22 = zero_zero_int )
         => ( A32 != product_Pair_int_int(zero_zero_int,A1) ) )
       => ( ! [Q5: int] :
              ( ( A32 = product_Pair_int_int(Q5,zero_zero_int) )
             => ( ( A22 != zero_zero_int )
               => ( A1 != aa_int_int(times_times_int(Q5),A22) ) ) )
         => ~ ! [R2: int,Q5: int] :
                ( ( A32 = product_Pair_int_int(Q5,R2) )
               => ( ( aa_int_int(sgn_sgn_int,R2) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R2)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q5),A22)),R2) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_9531_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A1,A22),A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q4,zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(times_times_int(Q4),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q4,R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4))
            & ( K3 = aa_int_int(plus_plus_int(aa_int_int(times_times_int(Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_9532_pi__half,axiom,
    aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = the_real(aTP_Lamm_kd) ).

% pi_half
tff(fact_9533_pi__def,axiom,
    pi = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),the_real(aTP_Lamm_kd)) ).

% pi_def
tff(fact_9534_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb)),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M)))))) ) ).

% divide_int_unfold
tff(fact_9535_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Nb: nat] :
      aa_int_int(modulo_modulo_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,M)),
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))),aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,L)),aa_int_int(minus_minus_int(aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),M))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),Nb))))) ) ).

% modulo_int_unfold
tff(fact_9536_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_9537_sgn__div__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2) != zero_zero_int )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) = aa_int_int(sgn_sgn_int,aa_int_int(times_times_int(A2),B2)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_9538_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(minus_minus_int(aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K)),aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_9539_sgn__le__0__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,Xc)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_9540_zero__le__sgn__iff,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,Xc))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc) ) ).

% zero_le_sgn_iff
tff(fact_9541_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri631733984087533419it_int(Nb),K))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_9542_signed__take__bit__negative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),zero_zero_int)
    <=> aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% signed_take_bit_negative_iff
tff(fact_9543_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_9544_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_nat_nat(suc,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_9545_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_9546_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: num] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,W)))),aa_num_nat(numeral_numeral_nat,Nb))
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_num_nat(pred_numeral,Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_9547_bin__nth__minus__Bit0,axiom,
    ! [Nb: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))),Nb)
      <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% bin_nth_minus_Bit0
tff(fact_9548_bin__nth__minus__Bit1,axiom,
    ! [Nb: nat,W: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,W))),Nb)
      <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,W)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% bin_nth_minus_Bit1
tff(fact_9549_real__sgn__eq,axiom,
    ! [Xc: real] : aa_real_real(sgn_sgn_real,Xc) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xc),aa_real_real(abs_abs_real,Xc)) ).

% real_sgn_eq
tff(fact_9550_bit__and__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se725231765392027082nd_int(K),L)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_and_int_iff
tff(fact_9551_sgn__root,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)) = aa_real_real(sgn_sgn_real,Xc) ) ) ).

% sgn_root
tff(fact_9552_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,K)),one_one_int)),Nb)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% bit_not_int_iff'
tff(fact_9553_sgn__eq,axiom,
    ! [Z: complex] : aa_complex_complex(sgn_sgn_complex,Z) = aa_complex_complex(divide1717551699836669952omplex(Z),real_V4546457046886955230omplex(aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% sgn_eq
tff(fact_9554_sgn__real__def,axiom,
    ! [A2: real] :
      aa_real_real(sgn_sgn_real,A2) = $ite(
        A2 = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A2),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_real_def
tff(fact_9555_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,M: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
     => ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(bit_se2923211474154528505it_int(M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_9556_sgn__integer__code,axiom,
    ! [K: code_integer] :
      sgn_sgn_Code_integer(K) = $ite(
        K = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),uminus1351360451143612070nteger(one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_9557_sgn__power__injE,axiom,
    ! [A2: real,Nb: nat,Xc: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A2)),Nb)) = Xc )
     => ( ( Xc = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B2)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_9558_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),M2)
              <=> aa_nat_o(bit_se1146084159140164899it_int(K),N) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))
              <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),N) ) ) ) ).

% int_bit_bound
tff(fact_9559_sgn__power__root,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc))),Nb)) = Xc ) ) ).

% sgn_power_root
tff(fact_9560_root__sgn__power,axiom,
    ! [Nb: nat,Ya: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Ya)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Ya)),Nb))) = Ya ) ) ).

% root_sgn_power
tff(fact_9561_bit__int__def,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% bit_int_def
tff(fact_9562_cis__Arg__unique,axiom,
    ! [Z: complex,Xc: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Z) = cis(Xc) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xc)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),pi)
         => ( aa_complex_real(arg,Z) = Xc ) ) ) ) ).

% cis_Arg_unique
tff(fact_9563_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,Xc: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ! [Y4: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y4)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y4)),Nb)) = Xc )
             => aa_real_o(P,Y4) ) ) ) ) ).

% split_root
tff(fact_9564_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Z) = cis(aa_complex_real(arg,Z)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Z))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Z)),pi) ) ) ).

% Arg_correct
tff(fact_9565_Bit__Operations_Oset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(Nb),K) = aa_int_int(plus_plus_int(K),aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% Bit_Operations.set_bit_eq
tff(fact_9566_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(Nb),K) = aa_int_int(minus_minus_int(K),aa_int_int(times_times_int(aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ).

% unset_bit_eq
tff(fact_9567_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K) = aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_9568_arctan__inverse,axiom,
    ! [Xc: real] :
      ( ( Xc != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xc)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Xc)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,Xc)) ) ) ).

% arctan_inverse
tff(fact_9569_num_Osize__gen_I3_J,axiom,
    ! [X33: num] : aa_num_nat(size_num,aa_num_num(bit1,X33)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X33)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(3)
tff(fact_9570_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ).

% mask_nat_positive_iff
tff(fact_9571_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_9572_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_Suc_0_iff
tff(fact_9573_nat__mask__eq,axiom,
    ! [Nb: nat] : aa_int_nat(nat2,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) ).

% nat_mask_eq
tff(fact_9574_less__eq__mask,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ).

% less_eq_mask
tff(fact_9575_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% mask_nonnegative_int
tff(fact_9576_not__mask__negative__int,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),zero_zero_int) ).

% not_mask_negative_int
tff(fact_9577_not__bit__Suc__0__numeral,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_9578_less__mask,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) ) ).

% less_mask
tff(fact_9579_bit__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_int_nat(nat2,K)),Nb)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ) ).

% bit_nat_iff
tff(fact_9580_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(plus_plus_int(K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_9581_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_9582_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(suc,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_mask_eq_exp
tff(fact_9583_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% mask_nat_less_exp
tff(fact_9584_bit__nat__def,axiom,
    ! [M: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,M),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% bit_nat_def
tff(fact_9585_mask__half__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ).

% mask_half_int
tff(fact_9586_mask__int__def,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = aa_int_int(minus_minus_int(aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) ).

% mask_int_def
tff(fact_9587_mask__nat__def,axiom,
    ! [Nb: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat) ).

% mask_nat_def
tff(fact_9588_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_int_int(plus_plus_int(K),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_9589_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : aa_num_nat(size_num,aa_num_num(bit0,X22)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(size_num,X22)),aa_nat_nat(suc,zero_zero_nat)) ).

% num.size_gen(2)
tff(fact_9590_root__def,axiom,
    ! [Nb: nat,Xc: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),Xc) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_ke,Nb)),Xc)) ).

% root_def
tff(fact_9591_Arg__def,axiom,
    ! [Z: complex] :
      aa_complex_real(arg,Z) = $ite(Z = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_kf(Z))) ).

% Arg_def
tff(fact_9592_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% not_negative_int_iff
tff(fact_9593_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% not_nonnegative_int_iff
tff(fact_9594_bit__not__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),Nb)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% bit_not_int_iff
tff(fact_9595_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(minus_minus_int(aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_9596_and__not__numerals_I1_J,axiom,
    aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = zero_zero_int ).

% and_not_numerals(1)
tff(fact_9597_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_9598_even__not__iff__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,K))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) ) ).

% even_not_iff_int
tff(fact_9599_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(4)
tff(fact_9600_and__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_9601_bit__minus__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(uminus_uminus_int,K)),Nb)
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(minus_minus_int(K),one_one_int))),Nb) ) ).

% bit_minus_int_iff
tff(fact_9602_and__not__numerals_I5_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(5)
tff(fact_9603_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,M)) ).

% and_not_numerals(7)
tff(fact_9604_and__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_9605_and__not__numerals_I9_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(9)
tff(fact_9606_and__not__numerals_I6_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(6)
tff(fact_9607_and__not__numerals_I8_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_9608_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_9609_bitNOT__integer__code,axiom,
    ! [I: code_integer] : bit_ri7632146776885996613nteger(I) = minus_8373710615458151222nteger(uminus1351360451143612070nteger(I),one_one_Code_integer) ).

% bitNOT_integer_code
tff(fact_9610_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K),L) = $ite(
        K = aa_int_int(uminus_uminus_int,one_one_int),
        aa_int_int(bit_ri7919022796975470100ot_int,L),
        $ite(
          L = aa_int_int(uminus_uminus_int,one_one_int),
          aa_int_int(bit_ri7919022796975470100ot_int,K),
          $ite(
            K = zero_zero_int,
            L,
            $ite(L = zero_zero_int,K,aa_int_int(plus_plus_int(aa_int_int(abs_abs_int,aa_int_int(minus_minus_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_9611_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_9612_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(K),L)),zero_zero_int)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% or_negative_int_iff
tff(fact_9613_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_9614_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(K),L)),zero_zero_int)
    <=> ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ).

% xor_negative_int_iff
tff(fact_9615_or__minus__numerals_I6_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(6)
tff(fact_9616_or__minus__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))) ).

% or_minus_numerals(2)
tff(fact_9617_or__nat__numerals_I2_J,axiom,
    ! [Ya: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ya))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ya)) ).

% or_nat_numerals(2)
tff(fact_9618_or__nat__numerals_I4_J,axiom,
    ! [Xc: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xc))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xc)) ).

% or_nat_numerals(4)
tff(fact_9619_or__nat__numerals_I1_J,axiom,
    ! [Ya: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ya))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ya)) ).

% or_nat_numerals(1)
tff(fact_9620_or__nat__numerals_I3_J,axiom,
    ! [Xc: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xc))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xc)) ).

% or_nat_numerals(3)
tff(fact_9621_and__minus__minus__numerals,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se1409905431419307370or_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9622_or__minus__minus__numerals,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,M))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,M)),one_one_int)),aa_int_int(minus_minus_int(aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9623_xor__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se725231765392027082nd_int(K),aa_int_int(bit_ri7919022796975470100ot_int,L))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),L)) ).

% xor_int_def
tff(fact_9624_bit__or__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se1409905431419307370or_int(K),L)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
        | aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_or_int_iff
tff(fact_9625_bit__xor__int__iff,axiom,
    ! [K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se6526347334894502574or_int(K),L)),Nb)
    <=> ~ ( aa_nat_o(bit_se1146084159140164899it_int(K),Nb)
        <=> aa_nat_o(bit_se1146084159140164899it_int(L),Nb) ) ) ).

% bit_xor_int_iff
tff(fact_9626_plus__and__or,axiom,
    ! [Xc: int,Ya: int] : aa_int_int(plus_plus_int(aa_int_int(bit_se725231765392027082nd_int(Xc),Ya)),aa_int_int(bit_se1409905431419307370or_int(Xc),Ya)) = aa_int_int(plus_plus_int(Xc),Ya) ).

% plus_and_or
tff(fact_9627_or__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Nb) = aa_int_nat(nat2,aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% or_nat_def
tff(fact_9628_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_int_int(bit_se1409905431419307370or_int(K),L)) ) ).

% or_greater_eq
tff(fact_9629_OR__lower,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(Xc),Ya)) ) ) ).

% OR_lower
tff(fact_9630_XOR__lower,axiom,
    ! [Xc: int,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Ya)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(Xc),Ya)) ) ) ).

% XOR_lower
tff(fact_9631_or__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,K)),aa_int_int(bit_ri7919022796975470100ot_int,L))) ).

% or_int_def
tff(fact_9632_or__not__numerals_I1_J,axiom,
    aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(1)
tff(fact_9633_or__not__numerals_I2_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(2)
tff(fact_9634_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9635_or__not__numerals_I3_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))) ).

% or_not_numerals(3)
tff(fact_9636_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9637_or__not__numerals_I6_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% or_not_numerals(6)
tff(fact_9638_XOR__upper,axiom,
    ! [Xc: int,Nb: nat,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(Xc),Ya)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_9639_OR__upper,axiom,
    ! [Xc: int,Nb: nat,Ya: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xc)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xc),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ya),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(Xc),Ya)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% OR_upper
tff(fact_9640_or__not__numerals_I5_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_9641_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% or_Suc_0_eq
tff(fact_9642_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_or_eq
tff(fact_9643_or__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9644_or__not__numerals_I9_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_9645_or__not__numerals_I8_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,M))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(plus_plus_int(one_one_int),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_9646_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_9647_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_9648_or__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,M),Nb) = $ite(
        M = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9649_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K),L) = $ite(
        ( ( K = aa_int_int(uminus_uminus_int,one_one_int) )
        | ( L = aa_int_int(uminus_uminus_int,one_one_int) ) ),
        aa_int_int(uminus_uminus_int,one_one_int),
        $ite(
          K = zero_zero_int,
          L,
          $ite(L = zero_zero_int,K,aa_int_int(plus_plus_int(aa_int_int(ord_max_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(modulo_modulo_int(L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% or_int_unfold
tff(fact_9650_Bit__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xc: $o] : bits_Bit_integer(code_integer_of_int(Xaa),(Xc)) = code_integer_of_int(aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,(Xc))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa))) ).

% Bit_integer.abs_eq
tff(fact_9651_or__minus__numerals_I5_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(5)
tff(fact_9652_or__minus__numerals_I1_J,axiom,
    ! [Nb: num] : aa_int_int(bit_se1409905431419307370or_int(one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(1)
tff(fact_9653_xor__nat__numerals_I4_J,axiom,
    ! [Xc: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xc))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xc)) ).

% xor_nat_numerals(4)
tff(fact_9654_xor__nat__numerals_I3_J,axiom,
    ! [Xc: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xc))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xc)) ).

% xor_nat_numerals(3)
tff(fact_9655_xor__nat__numerals_I2_J,axiom,
    ! [Ya: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ya))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ya)) ).

% xor_nat_numerals(2)
tff(fact_9656_xor__nat__numerals_I1_J,axiom,
    ! [Ya: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Ya))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Ya)) ).

% xor_nat_numerals(1)
tff(fact_9657_or__minus__numerals_I4_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(4)
tff(fact_9658_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(8)
tff(fact_9659_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,M: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,M)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(7)
tff(fact_9660_or__minus__numerals_I3_J,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(3)
tff(fact_9661_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(bit_or_not_num_neg(one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9662_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Nb)),one) = aa_num_num(bit0,one) ).

% or_not_num_neg.simps(4)
tff(fact_9663_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,M: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Nb)),aa_num_num(bit1,M)) = aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(Nb),M)) ).

% or_not_num_neg.simps(6)
tff(fact_9664_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9665_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit1,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_9666_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,M: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Nb)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),M)) ).

% or_not_num_neg.simps(5)
tff(fact_9667_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Nb: num,M: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),aa_num_num(bit1,M)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),M)) ).

% or_not_num_neg.simps(9)
tff(fact_9668_xor__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Nb) = aa_int_nat(nat2,aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,M)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% xor_nat_def
tff(fact_9669_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit0,M)) = aa_num_num(bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_9670_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,M: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),aa_num_num(bit0,M)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),M)) ).

% or_not_num_neg.simps(8)
tff(fact_9671_or__not__num__neg_Oelims,axiom,
    ! [Xc: num,Xaa: num,Ya: num] :
      ( ( aa_num_num(bit_or_not_num_neg(Xc),Xaa) = Ya )
     => ( ( ( Xc = one )
         => ( ( Xaa = one )
           => ( Ya != one ) ) )
       => ( ( ( Xc = one )
           => ! [M4: num] :
                ( ( Xaa = aa_num_num(bit0,M4) )
               => ( Ya != aa_num_num(bit1,M4) ) ) )
         => ( ( ( Xc = one )
             => ! [M4: num] :
                  ( ( Xaa = aa_num_num(bit1,M4) )
                 => ( Ya != aa_num_num(bit1,M4) ) ) )
           => ( ( ? [N: num] : Xc = aa_num_num(bit0,N)
               => ( ( Xaa = one )
                 => ( Ya != aa_num_num(bit0,one) ) ) )
             => ( ! [N: num] :
                    ( ( Xc = aa_num_num(bit0,N) )
                   => ! [M4: num] :
                        ( ( Xaa = aa_num_num(bit0,M4) )
                       => ( Ya != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M4)) ) ) )
               => ( ! [N: num] :
                      ( ( Xc = aa_num_num(bit0,N) )
                     => ! [M4: num] :
                          ( ( Xaa = aa_num_num(bit1,M4) )
                         => ( Ya != aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(N),M4)) ) ) )
                 => ( ( ? [N: num] : Xc = aa_num_num(bit1,N)
                     => ( ( Xaa = one )
                       => ( Ya != one ) ) )
                   => ( ! [N: num] :
                          ( ( Xc = aa_num_num(bit1,N) )
                         => ! [M4: num] :
                              ( ( Xaa = aa_num_num(bit0,M4) )
                             => ( Ya != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M4)) ) ) )
                     => ~ ! [N: num] :
                            ( ( Xc = aa_num_num(bit1,N) )
                           => ! [M4: num] :
                                ( ( Xaa = aa_num_num(bit1,M4) )
                               => ( Ya != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N),M4)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9672_numeral__or__not__num__eq,axiom,
    ! [M: num,Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),Nb)) = aa_int_int(uminus_uminus_int,aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% numeral_or_not_num_eq
tff(fact_9673_int__numeral__not__or__num__neg,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,M))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Nb),M))) ).

% int_numeral_not_or_num_neg
tff(fact_9674_int__numeral__or__not__num__neg,axiom,
    ! [M: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,M)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(M),Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_9675_xor__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Nb) = $ite(
        M = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,M,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_9676_xor__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          aa_o_nat(zero_n2687167440665602831ol_nat,
            ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_9677_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_xor_eq
tff(fact_9678_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_o_nat(zero_n2687167440665602831ol_nat,aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_o_nat(zero_n2687167440665602831ol_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% xor_Suc_0_eq
tff(fact_9679_int__lsb__numeral_I6_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) ).

% int_lsb_numeral(6)
tff(fact_9680_int__lsb__numeral_I3_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_num_int(numeral_numeral_int,one)) ).

% int_lsb_numeral(3)
tff(fact_9681_int__lsb__numeral_I8_J,axiom,
    ! [W: num] : ~ aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,W)))) ).

% int_lsb_numeral(8)
tff(fact_9682_int__lsb__numeral_I5_J,axiom,
    aa_int_o(least_4859182151741483524sb_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) ).

% int_lsb_numeral(5)
tff(fact_9683_lsb__integer__code,axiom,
    ! [Xc: code_integer] :
      ( aa_Code_integer_o(least_7544222001954398261nteger,Xc)
    <=> aa_nat_o(bit_se9216721137139052372nteger(Xc),zero_zero_nat) ) ).

% lsb_integer_code
tff(fact_9684_lsb__int__def,axiom,
    ! [I: int] :
      ( aa_int_o(least_4859182151741483524sb_int,I)
    <=> aa_nat_o(bit_se1146084159140164899it_int(I),zero_zero_nat) ) ).

% lsb_int_def
tff(fact_9685_bin__last__conv__lsb,axiom,
    aTP_Lamm_kg = least_4859182151741483524sb_int ).

% bin_last_conv_lsb
tff(fact_9686_dup__1,axiom,
    code_dup(one_one_Code_integer) = numera6620942414471956472nteger(aa_num_num(bit0,one)) ).

% dup_1
tff(fact_9687_pow_Osimps_I1_J,axiom,
    ! [Xc: num] : aa_num_num(pow(Xc),one) = Xc ).

% pow.simps(1)
tff(fact_9688_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [Xc: nat,Ya: int] :
      ( ( aa_nat_int(vEBT_V9176841429113362141ildupi,Xc) = Ya )
     => ( aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),Xc)
       => ( ( ( Xc = zero_zero_nat )
           => ( ( Ya = one_one_int )
             => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),zero_zero_nat) ) )
         => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Ya = one_one_int )
               => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Ya = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,one))),aa_int_int(plus_plus_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_nat_int(vEBT_V9176841429113362141ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V3352910403632780892pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_9689_vebt__buildup_Opelims,axiom,
    ! [Xc: nat,Ya: vEBT_VEBT] :
      ( ( aa_nat_VEBT_VEBT(vEBT_vebt_buildup,Xc) = Ya )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xc)
       => ( ( ( Xc = zero_zero_nat )
           => ( ( Ya = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Ya = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Ya = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half)) ),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2)),replicate_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half)),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,half))),aa_nat_VEBT_VEBT(vEBT_vebt_buildup,aa_nat_nat(suc,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_9690_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_V8646137997579335489_i_l_d,Xc) = Ya )
     => ( aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),Xc)
       => ( ( ( Xc = zero_zero_nat )
           => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
             => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),zero_zero_nat) ) )
         => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Ya = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit1,one))))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(vEBT_V8646137997579335489_i_l_d,half))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V5144397997797733112_d_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_9691_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_V8346862874174094_d_u_p,Xc) = Ya )
     => ( aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),Xc)
       => ( ( ( Xc = zero_zero_nat )
           => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),zero_zero_nat) ) )
         => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,Va2)) )
                 => ( ( Ya = $ite(
                          aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,Va2))),
                          aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                            $let(
                              half: nat,
                              half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                              aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,half))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),half)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) )),
                          $let(
                            half: nat,
                            half:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                            aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_nat_nat(vEBT_V8346862874174094_d_u_p,aa_nat_nat(suc,half)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,half))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V8346862874174094_d_u_p,half)),one_one_nat))) ) ) )
                   => ~ aa_nat_o(accp_nat(vEBT_V1247956027447740395_p_rel),aa_nat_nat(suc,aa_nat_nat(suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_9692_VEBT__internal_OTb_Opelims,axiom,
    ! [Xc: nat,Ya: int] :
      ( ( aa_nat_int(vEBT_VEBT_Tb,Xc) = Ya )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),Xc)
       => ( ( ( Xc = zero_zero_nat )
           => ( ( Ya = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),zero_zero_nat) ) )
         => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Ya = aa_num_int(numeral_numeral_int,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Ya = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_int_int(plus_plus_int(aa_int_int(plus_plus_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_int_int(times_times_int(aa_nat_int(vEBT_VEBT_Tb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel2),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_9693_VEBT__internal_OTb_H_Opelims,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_VEBT_Tb2,Xc) = Ya )
     => ( aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),Xc)
       => ( ( ( Xc = zero_zero_nat )
           => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
             => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),zero_zero_nat) ) )
         => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Ya = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_VEBT_Tb2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_VEBT_Tb_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_9694_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xc: nat,Ya: nat] :
      ( ( aa_nat_nat(vEBT_V441764108873111860ildupi,Xc) = Ya )
     => ( aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),Xc)
       => ( ( ( Xc = zero_zero_nat )
           => ( ( Ya = aa_nat_nat(suc,zero_zero_nat) )
             => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),zero_zero_nat) ) )
         => ( ( ( Xc = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Ya = aa_nat_nat(suc,zero_zero_nat) )
               => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [N: nat] :
                  ( ( Xc = aa_nat_nat(suc,aa_nat_nat(suc,N)) )
                 => ( ( Ya = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(vEBT_V441764108873111860ildupi,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))))) )
                   => ~ aa_nat_o(accp_nat(vEBT_V2957053500504383685pi_rel),aa_nat_nat(suc,aa_nat_nat(suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_9695_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( cis(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_9696_sin__times__pi__eq__0,axiom,
    ! [Xc: real] :
      ( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xc),pi)) = zero_zero_real )
    <=> aa_set_real_o(member_real(Xc),ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_9697_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_9698_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( aa_set_real_o(member_real(Nb),ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_9699_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space,Xc) = Ya )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),Xc)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary) )
               => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,aa_num_num(bit0,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space,Summary))),size_s6755466524823107622T_VEBT(TreeList2))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_9700_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Xc) = Ya )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),Xc)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary) )
               => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit1,one)))),aa_VEBT_VEBT_nat(vEBT_VEBT_space2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_space2,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_space_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_9701_VEBT__internal_Ocnt_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: real] :
      ( ( aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Xc) = Ya )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),Xc)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = one_one_real )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary) )
               => ( ( Ya = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_VEBT_VEBT_real(vEBT_VEBT_cnt,Summary))),aa_real_real(foldr_real_real(plus_plus_real,map_VEBT_VEBT_real(vEBT_VEBT_cnt,TreeList2)),zero_zero_real)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel2),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_9702_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Xc) = Ya )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),Xc)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = one_one_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary) )
               => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_VEBT_VEBT_nat(vEBT_VEBT_cnt2,Summary))),aa_nat_nat(foldr_nat_nat(plus_plus_nat,map_VEBT_VEBT_nat(vEBT_VEBT_cnt2,TreeList2)),zero_zero_nat)) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_cnt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Info2,Deg2,TreeList2),Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_9703_vebt__maxt_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: option_nat] :
      ( ( vEBT_vebt_maxt(Xc) = Ya )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),Xc)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = $ite(
                      (B3),
                      some_nat(one_one_nat),
                      $ite((A3),some_nat(zero_zero_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2) )
               => ( ( Ya = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz) )
                 => ( ( Ya = some_nat(Ma) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_9704_vebt__mint_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: option_nat] :
      ( ( vEBT_vebt_mint(Xc) = Ya )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),Xc)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = $ite(
                      (A3),
                      some_nat(zero_zero_nat),
                      $ite((B3),some_nat(one_one_nat),none_nat) ) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2) )
               => ( ( Ya = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz) )
                 => ( ( Ya = some_nat(Mi) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_9705_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_t,Xc) = Ya )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),Xc)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((A3),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2) )
               => ( ( Ya = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz) )
                 => ( ( Ya = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_i_n_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_9706_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_a_x_t,Xc) = Ya )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),Xc)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),
                      $ite((B3),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat))) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2) )
               => ( ( Ya = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz) )
                 => ( ( Ya = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T_m_a_x_t_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_9707_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_T_m_i_n_N_u_l_l,Xc) = Ya )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),Xc)
       => ( ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) )
           => ( ( Ya = one_one_nat )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_o_VEBT_VEBT(vEBT_Leaf($false),$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf($true),(Uv)) )
               => ( ( Ya = one_one_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_o_VEBT_VEBT(vEBT_Leaf($true),(Uv))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),$true) )
                 => ( ( Ya = one_one_nat )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),$true)) ) )
             => ( ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux),Uy) )
                   => ( ( Ya = one_one_nat )
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux),Uy)) ) )
               => ~ ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc) )
                     => ( ( Ya = one_one_nat )
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_T5462971552011256508_l_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_9708_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xc: vEBT_VEBT,Ya: $o] :
      ( ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xc)
      <=> (Ya) )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xc)
       => ( ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) )
           => ( (Ya)
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_o_VEBT_VEBT(vEBT_Leaf($false),$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf($true),(Uv)) )
               => ( ~ (Ya)
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_o_VEBT_VEBT(vEBT_Leaf($true),(Uv))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),$true) )
                 => ( ~ (Ya)
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),$true)) ) )
             => ( ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux),Uy) )
                   => ( (Ya)
                     => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux),Uy)) ) )
               => ~ ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc) )
                     => ( ~ (Ya)
                       => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_9709_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xc: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xc)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xc)
       => ( ! [Uv: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf($true),(Uv)) )
             => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_o_VEBT_VEBT(vEBT_Leaf($true),(Uv))) )
         => ( ! [Uu2: $o] :
                ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),$true) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_o_VEBT_VEBT(vEBT_Leaf((Uu2)),$true)) )
           => ~ ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                  ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_9710_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xc: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Xc)
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),Xc)
       => ( ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf($false),$false) )
           => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_o_VEBT_VEBT(vEBT_Leaf($false),$false)) )
         => ~ ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux),Uy) )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_V6963167321098673237ll_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(none_P5556105721700978146at_nat,Uw2,Ux),Uy)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_9711_setceilmax,axiom,
    ! [S2: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,Nb: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,S2),M)
     => ( ! [X3: vEBT_VEBT] :
            ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Listy))
           => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,X3),Nb) )
       => ( ( M = aa_nat_nat(suc,Nb) )
         => ( ! [X3: vEBT_VEBT] :
                ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(X3),set_VEBT_VEBT2(Listy))
               => ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X3)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) )
           => ( ( aa_nat_int(semiri1314217659103216013at_int,aa_VEBT_VEBT_nat(vEBT_VEBT_height,S2)) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) )
             => ( aa_nat_int(semiri1314217659103216013at_int,lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(S2,set_VEBT_VEBT2(Listy))))) = aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_9712_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,I)
    <=> ( modulo364778990260209775nteger(I,numera6620942414471956472nteger(aa_num_num(bit0,one))) != zero_z3403309356797280102nteger ) ) ).

% bin_last_integer_nbe
tff(fact_9713_height__compose__list,axiom,
    ! [Ta: vEBT_VEBT,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Ta),set_VEBT_VEBT2(TreeLista))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,Ta)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summarya,set_VEBT_VEBT2(TreeLista))))) ) ).

% height_compose_list
tff(fact_9714_max__ins__scaled,axiom,
    ! [Nb: nat,X14a: vEBT_VEBT,M: nat,X13a: list_VEBT_VEBT] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14a))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),lattic8265883725875713057ax_nat(insert_nat(aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14a),image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13a))))))) ).

% max_ins_scaled
tff(fact_9715_height__i__max,axiom,
    ! [I: nat,X13a: list_VEBT_VEBT,Foo: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13a))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(X13a),I))),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Foo),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13a))))) ) ).

% height_i_max
tff(fact_9716_max__idx__list,axiom,
    ! [I: nat,X13a: list_VEBT_VEBT,Nb: nat,X14a: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_s6755466524823107622T_VEBT(X13a))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(X13a),I)))),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_VEBT_VEBT_nat(vEBT_VEBT_height,X14a)),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,set_VEBT_VEBT2(X13a)))))))) ) ).

% max_idx_list
tff(fact_9717_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ( lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ed,Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_9718_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_VEBT_height,aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu,Deg,TreeLista),Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summarya,set_VEBT_VEBT2(TreeLista))))) ).

% VEBT_internal.height.simps(2)
tff(fact_9719_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xc) = Ya )
     => ( ( ? [A3: $o,B3: $o] : Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))
         => ( Ya != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary) )
             => ( Ya != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_9720_divide__nat__def,axiom,
    ! [M: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_kh(M),Nb)))) ).

% divide_nat_def
tff(fact_9721_bin__last__integer_Oabs__eq,axiom,
    ! [Xc: int] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,code_integer_of_int(Xc))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xc) ) ).

% bin_last_integer.abs_eq
tff(fact_9722_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xc: vEBT_VEBT,Ya: nat] :
      ( ( aa_VEBT_VEBT_nat(vEBT_VEBT_height,Xc) = Ya )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),Xc)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3)) )
             => ( ( Ya = zero_zero_nat )
               => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),aa_o_VEBT_VEBT(vEBT_Leaf((A3)),(B3))) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xc = aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary) )
               => ( ( Ya = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),lattic8265883725875713057ax_nat(image_VEBT_VEBT_nat(vEBT_VEBT_height,insert_VEBT_VEBT(Summary,set_VEBT_VEBT2(TreeList2))))) )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_VEBT_height_rel),aa_VEB5832953974749077948T_VEBT(vEBT_Node(Uu2,Deg2,TreeList2),Summary)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_9723_bij__betw__Suc,axiom,
    ! [M3: set_nat,N5: set_nat] :
      ( aa_set_nat_o(bij_betw_nat_nat(suc,M3),N5)
    <=> ( image_nat_nat(suc,M3) = N5 ) ) ).

% bij_betw_Suc
tff(fact_9724_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J2: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J2)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J2)) ).

% image_Suc_atLeastLessThan
tff(fact_9725_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J2: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J2)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J2)) ).

% image_Suc_atLeastAtMost
tff(fact_9726_range__mult,axiom,
    ! [A2: real] :
      image_real_real(aa_rea7340605660300375871l_real(times_times_real,A2),top_top_set_real) = $ite(A2 = zero_zero_real,insert_real(zero_zero_real,bot_bot_set_real),top_top_set_real) ).

% range_mult
tff(fact_9727_zero__notin__Suc__image,axiom,
    ! [A: set_nat] : ~ aa_set_nat_o(member_nat(zero_zero_nat),image_nat_nat(suc,A)) ).

% zero_notin_Suc_image
tff(fact_9728_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B2: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(A2),B2)) = set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,A2),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% image_int_atLeastAtMost
tff(fact_9729_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B2: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or4665077453230672383an_nat(A2),B2)) = set_or4662586982721622107an_int(aa_nat_int(semiri1314217659103216013at_int,A2),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% image_int_atLeastLessThan
tff(fact_9730_image__Suc__lessThan,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb) ).

% image_Suc_lessThan
tff(fact_9731_image__Suc__atMost,axiom,
    ! [Nb: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,Nb)) ).

% image_Suc_atMost
tff(fact_9732_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_9733_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_9734_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_9735_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,Nb)) = insert_nat(zero_zero_nat,image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_9736_range__mod,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ki,Nb),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb) ) ) ).

% range_mod
tff(fact_9737_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_kj(L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(minus_minus_int(U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9738_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : image_4470545334726330049nteger(aTP_Lamm_kk(L),set_or8404916559141939852nteger(zero_z3403309356797280102nteger,minus_8373710615458151222nteger(U,L))) = set_or8404916559141939852nteger(L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_9739_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
     => ( set_or4662586982721622107an_int(zero_zero_int,U) = image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_ord_lessThan_nat,aa_int_nat(nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_9740_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xc: nat,Ya: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kl,C2),aa_nat_set_nat(set_or4665077453230672383an_nat(Xc),Ya)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C2),Ya),
        aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xc),C2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Ya),C2)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xc),Ya),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9741_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_9742_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se8568078237143864401it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_9743_drop__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se8568078237143864401it_int(Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% drop_bit_negative_int_iff
tff(fact_9744_drop__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% drop_bit_minus_one
tff(fact_9745_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_9746_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9747_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_9748_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_9749_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,K)))) = aa_int_int(bit_se8568078237143864401it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_9750_drop__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se8568078237143864401it_int(Nb),K)) ).

% drop_bit_nat_eq
tff(fact_9751_shiftr__integer__conv__div__pow2,axiom,
    ! [Nb: nat,Xc: code_integer] : bit_se3928097537394005634nteger(Nb,Xc) = divide6298287555418463151nteger(Xc,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) ).

% shiftr_integer_conv_div_pow2
tff(fact_9752_drop__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_int_def
tff(fact_9753_drop__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),M) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_nat_def
tff(fact_9754_bin__rest__code,axiom,
    ! [I: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,I),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(bit_se8568078237143864401it_int(one_one_nat),I) ).

% bin_rest_code
tff(fact_9755_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se545348938243370406it_int(Nb),K))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_9756_push__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se545348938243370406it_int(Nb),K)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% push_bit_negative_int_iff
tff(fact_9757_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% push_bit_of_Suc_0
tff(fact_9758_drop__bit__push__bit__int,axiom,
    ! [M: nat,Nb: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(M),aa_int_int(bit_se545348938243370406it_int(Nb),K)) = aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_9759_flip__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_9760_set__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9761_push__bit__nat__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_int_nat(nat2,K)) = aa_int_nat(nat2,aa_int_int(bit_se545348938243370406it_int(Nb),K)) ).

% push_bit_nat_eq
tff(fact_9762_push__bit__int__code_I1_J,axiom,
    ! [I: int] : aa_int_int(bit_se545348938243370406it_int(zero_zero_nat),I) = I ).

% push_bit_int_code(1)
tff(fact_9763_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(M),K)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_9764_Bit__Operations_Oset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se7879613467334960850it_int(Nb),K) = aa_int_int(bit_se1409905431419307370or_int(K),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% Bit_Operations.set_bit_int_def
tff(fact_9765_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M),Q3)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q3),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_9766_drop__bit__int__code_I1_J,axiom,
    ! [I: int] : aa_int_int(bit_se8568078237143864401it_int(zero_zero_nat),I) = I ).

% drop_bit_int_code(1)
tff(fact_9767_flip__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2159334234014336723it_int(Nb),K) = aa_int_int(bit_se6526347334894502574or_int(K),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% flip_bit_int_def
tff(fact_9768_shiftl__integer__conv__mult__pow2,axiom,
    ! [Nb: nat,Xc: code_integer] : bit_se7788150548672797655nteger(Nb,Xc) = times_3573771949741848930nteger(Xc,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_9769_unset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se4203085406695923979it_int(Nb),K) = aa_int_int(bit_se725231765392027082nd_int(K),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int))) ).

% unset_bit_int_def
tff(fact_9770_push__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(Nb),K) = aa_int_int(times_times_int(K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_int_def
tff(fact_9771_push__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),M) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% push_bit_nat_def
tff(fact_9772_push__bit__minus__one,axiom,
    ! [Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),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))),Nb)) ).

% push_bit_minus_one
tff(fact_9773_drop__bit__int__code_I2_J,axiom,
    ! [Nb: nat] : aa_int_int(bit_se8568078237143864401it_int(aa_nat_nat(suc,Nb)),zero_zero_int) = zero_zero_int ).

% drop_bit_int_code(2)
tff(fact_9774_bin__rest__integer_Oabs__eq,axiom,
    ! [Xc: int] : bits_b2549910563261871055nteger(code_integer_of_int(Xc)) = code_integer_of_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xc),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% bin_rest_integer.abs_eq
tff(fact_9775_bin__rest__integer__code,axiom,
    ! [I: code_integer] : bits_b2549910563261871055nteger(I) = divide6298287555418463151nteger(I,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% bin_rest_integer_code
tff(fact_9776_upto__aux__rec,axiom,
    ! [I: int,J2: int,Js: list_int] :
      upto_aux(I,J2,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J2),I),Js,upto_aux(I,aa_int_int(minus_minus_int(J2),one_one_int),cons_int(J2,Js))) ).

% upto_aux_rec
tff(fact_9777_floor__real__def,axiom,
    ! [Xc: real] : aa_real_int(archim6058952711729229775r_real,Xc) = the_int(aTP_Lamm_km(Xc)) ).

% floor_real_def
tff(fact_9778_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_concat_bit(aa_nat_nat(suc,Nb),K),L) = aa_int_int(plus_plus_int(aa_int_int(modulo_modulo_int(K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_concat_bit(Nb,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_9779_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_9780_concat__bit__of__zero__2,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_concat_bit(Nb,K),zero_zero_int) = aa_int_int(bit_se2923211474154528505it_int(Nb),K) ).

% concat_bit_of_zero_2
tff(fact_9781_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_concat_bit(Nb,K),L))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_9782_concat__bit__negative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_concat_bit(Nb,K),L)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ).

% concat_bit_negative_iff
tff(fact_9783_concat__bit__of__zero__1,axiom,
    ! [Nb: nat,L: int] : aa_int_int(bit_concat_bit(Nb,zero_zero_int),L) = aa_int_int(bit_se545348938243370406it_int(Nb),L) ).

% concat_bit_of_zero_1
tff(fact_9784_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,M: nat,L: int,R3: int] : aa_int_int(bit_concat_bit(Nb,K),aa_int_int(bit_concat_bit(M,L),R3)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M),Nb),aa_int_int(bit_concat_bit(Nb,K),L)),R3) ).

% concat_bit_assoc
tff(fact_9785_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K: int,L: int,R3: int,S2: int] :
      ( ( aa_int_int(bit_concat_bit(Nb,K),L) = aa_int_int(bit_concat_bit(Nb,R3),S2) )
    <=> ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_int_int(bit_se2923211474154528505it_int(Nb),R3) )
        & ( L = S2 ) ) ) ).

% concat_bit_eq_iff
tff(fact_9786_concat__bit__take__bit__eq,axiom,
    ! [Nb: nat,B2: int] : bit_concat_bit(Nb,aa_int_int(bit_se2923211474154528505it_int(Nb),B2)) = bit_concat_bit(Nb,B2) ).

% concat_bit_take_bit_eq
tff(fact_9787_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_concat_bit(Nb,K),L) = aa_int_int(plus_plus_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se545348938243370406it_int(Nb),L)) ).

% concat_bit_eq
tff(fact_9788_concat__bit__def,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_concat_bit(Nb,K),L) = aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se545348938243370406it_int(Nb),L)) ).

% concat_bit_def
tff(fact_9789_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_concat_bit(M,K),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M)
          & aa_nat_o(bit_se1146084159140164899it_int(K),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M),Nb)
          & aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_9790_signed__take__bit__eq__concat__bit,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(bit_concat_bit(Nb,K),aa_int_int(uminus_uminus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_9791_floor__rat__def,axiom,
    ! [Xc: rat] : aa_rat_int(archim3151403230148437115or_rat,Xc) = the_int(aTP_Lamm_kn(Xc)) ).

% floor_rat_def
tff(fact_9792_vebt__minti_Opelims,axiom,
    ! [Xc: vEBT_VEBTi,Ya: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_minti(Xc) = Ya )
     => ( aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),Xc)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBTi(vEBT_Leafi((A3)),(B3)) )
             => ( ( Ya = $ite(
                      (A3),
                      heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),
                      $ite((B3),heap_T3487192422709364219on_nat(some_nat(one_one_nat)),heap_T3487192422709364219on_nat(none_nat)) ) )
               => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),aa_o_VEBT_VEBTi(vEBT_Leafi((A3)),(B3))) ) )
         => ( ! [Uu2: nat,Uv: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw2) )
               => ( ( Ya = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                  ( ( Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz) )
                 => ( ( Ya = heap_T3487192422709364219on_nat(some_nat(Mi)) )
                   => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_minti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) ) ) ) ) ) ) ).

% vebt_minti.pelims
tff(fact_9793_less__eq__rat__def,axiom,
    ! [Xc: rat,Ya: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xc),Ya)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xc),Ya)
        | ( Xc = Ya ) ) ) ).

% less_eq_rat_def
tff(fact_9794_abs__rat__def,axiom,
    ! [A2: rat] :
      aa_rat_rat(abs_abs_rat,A2) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,A2),A2) ).

% abs_rat_def
tff(fact_9795_sgn__rat__def,axiom,
    ! [A2: rat] :
      aa_rat_rat(sgn_sgn_rat,A2) = $ite(
        A2 = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A2),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_rat_def
tff(fact_9796_obtain__pos__sum,axiom,
    ! [R3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
     => ~ ! [S3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S3)
           => ! [T5: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T5)
               => ( R3 != aa_rat_rat(plus_plus_rat(S3),T5) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9797_vebt__maxti_Opelims,axiom,
    ! [Xc: vEBT_VEBTi,Ya: heap_T2636463487746394924on_nat] :
      ( ( vEBT_vebt_maxti(Xc) = Ya )
     => ( aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),Xc)
       => ( ! [A3: $o,B3: $o] :
              ( ( Xc = aa_o_VEBT_VEBTi(vEBT_Leafi((A3)),(B3)) )
             => ( ( Ya = $ite(
                      (B3),
                      heap_T3487192422709364219on_nat(some_nat(one_one_nat)),
                      $ite((A3),heap_T3487192422709364219on_nat(some_nat(zero_zero_nat)),heap_T3487192422709364219on_nat(none_nat)) ) )
               => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),aa_o_VEBT_VEBTi(vEBT_Leafi((A3)),(B3))) ) )
         => ( ! [Uu2: nat,Uv: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw2) )
               => ( ( Ya = heap_T3487192422709364219on_nat(none_nat) )
                 => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uu2,Uv),Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                  ( ( Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz) )
                 => ( ( Ya = heap_T3487192422709364219on_nat(some_nat(Ma)) )
                   => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_vebt_maxti_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy),Uz)) ) ) ) ) ) ) ).

% vebt_maxti.pelims
tff(fact_9798_VEBT__internal_OminNulli_Opelims,axiom,
    ! [Xc: vEBT_VEBTi,Ya: heap_Time_Heap_o] :
      ( ( vEBT_VEBT_minNulli(Xc) = Ya )
     => ( aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),Xc)
       => ( ( ( Xc = aa_o_VEBT_VEBTi(vEBT_Leafi($false),$false) )
           => ( ( Ya = heap_Time_return_o($true) )
             => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),aa_o_VEBT_VEBTi(vEBT_Leafi($false),$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xc = aa_o_VEBT_VEBTi(vEBT_Leafi($true),(Uv)) )
               => ( ( Ya = heap_Time_return_o($false) )
                 => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),aa_o_VEBT_VEBTi(vEBT_Leafi($true),(Uv))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xc = aa_o_VEBT_VEBTi(vEBT_Leafi((Uu2)),$true) )
                 => ( ( Ya = heap_Time_return_o($false) )
                   => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),aa_o_VEBT_VEBTi(vEBT_Leafi((Uu2)),$true)) ) )
             => ( ! [Uw2: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] :
                    ( ( Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux),Uy) )
                   => ( ( Ya = heap_Time_return_o($true) )
                     => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(none_P5556105721700978146at_nat,Uw2,Ux),Uy)) ) )
               => ~ ! [Uz: product_prod_nat_nat,Va: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
                      ( ( Xc = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc) )
                     => ( ( Ya = heap_Time_return_o($false) )
                       => ~ aa_VEBT_VEBTi_o(accp_VEBT_VEBTi(vEBT_V5740978063120863272li_rel),aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(Uz),Va,Vb),Vc)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.pelims
tff(fact_9799_diff__rat__def,axiom,
    ! [Q3: rat,R3: rat] : aa_rat_rat(minus_minus_rat(Q3),R3) = aa_rat_rat(plus_plus_rat(Q3),aa_rat_rat(uminus_uminus_rat,R3)) ).

% diff_rat_def
tff(fact_9800_normalize__negative,axiom,
    ! [Q3: int,P3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Q3),zero_zero_int)
     => ( normalize(product_Pair_int_int(P3,Q3)) = normalize(product_Pair_int_int(aa_int_int(uminus_uminus_int,P3),aa_int_int(uminus_uminus_int,Q3))) ) ) ).

% normalize_negative
tff(fact_9801_normalize__denom__pos,axiom,
    ! [R3: product_prod_int_int,P3: int,Q3: int] :
      ( ( normalize(R3) = product_Pair_int_int(P3,Q3) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).

% normalize_denom_pos
tff(fact_9802_rat__minus__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P3),Q3)) = produc4245557441103728435nt_int(aTP_Lamm_kp(Q3),quotient_of(P3)) ).

% rat_minus_code
tff(fact_9803_quotient__of__div,axiom,
    ! [R3: rat,Nb: int,D2: int] :
      ( ( quotient_of(R3) = product_Pair_int_int(Nb,D2) )
     => ( R3 = aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,Nb)),aa_int_rat(ring_1_of_int_rat,D2)) ) ) ).

% quotient_of_div
tff(fact_9804_rat__floor__code,axiom,
    ! [P3: rat] : aa_rat_int(archim3151403230148437115or_rat,P3) = produc8211389475949308722nt_int(divide_divide_int,quotient_of(P3)) ).

% rat_floor_code
tff(fact_9805_quotient__of__denom__pos,axiom,
    ! [R3: rat,P3: int,Q3: int] :
      ( ( quotient_of(R3) = product_Pair_int_int(P3,Q3) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q3) ) ).

% quotient_of_denom_pos
tff(fact_9806_rat__less__code,axiom,
    ! [P3: rat,Q3: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P3),Q3)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_kr(Q3)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_9807_rat__divide__code,axiom,
    ! [P3: rat,Q3: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P3),Q3)) = produc4245557441103728435nt_int(aTP_Lamm_kt(Q3),quotient_of(P3)) ).

% rat_divide_code
tff(fact_9808_divide__rat__def,axiom,
    ! [Q3: rat,R3: rat] : aa_rat_rat(divide_divide_rat(Q3),R3) = aa_rat_rat(times_times_rat(Q3),aa_rat_rat(inverse_inverse_rat,R3)) ).

% divide_rat_def
tff(fact_9809_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : aa_int_o(aa_int_fun_int_o(ord_less_int,groups9116527308978886569_o_int(zero_n2684676970156552555ol_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)),Bs)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),size_size_list_o(Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_9810_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(one_one_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K)) ).

% Frct_code_post(5)
tff(fact_9811_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))) = aa_rat_rat(divide_divide_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_9812_Cauchy__iff2,axiom,
    ! [X: fun_nat_real] :
      ( topolo4055970368930404560y_real(X)
    <=> ! [J: nat] :
        ? [M9: nat] :
        ! [M8: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M8)
         => ! [N6: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N6)
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X,M8)),aa_nat_real(X,N6)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J)))) ) ) ) ).

% Cauchy_iff2
tff(fact_9813_sdiv__int__numeral__numeral,axiom,
    ! [M: num,Nb: num] : aa_int_int(signed6714573509424544716de_int(aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,M)),aa_num_int(numeral_numeral_int,Nb)) ).

% sdiv_int_numeral_numeral
tff(fact_9814_signed__divide__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(signed6714573509424544716de_int(K),L) = aa_int_int(times_times_int(aa_int_int(times_times_int(aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))) ).

% signed_divide_int_def
tff(fact_9815_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( fI_QUERY(P,Q,top_top_assn)
     => entails(P,aa_assn_assn(times_times_assn(Q),top_top_assn)) ) ).

% entails_solve_init(1)
tff(fact_9816_VEBT_Osize_I3_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT] : aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12,X13a),X14a)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(size_size_VEBT_VEBT,X13a)),aa_VEBT_VEBT_nat(size_size_VEBT_VEBT,X14a))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size(3)
tff(fact_9817_FI__QUERY__def,axiom,
    ! [P: assn,Q: assn,F: assn] :
      ( fI_QUERY(P,Q,F)
    <=> entails(P,aa_assn_assn(times_times_assn(Q),F)) ) ).

% FI_QUERY_def
tff(fact_9818_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F: assn] :
      ( fI_QUERY(P,Q,F)
     => entails(P,aa_assn_assn(times_times_assn(Q),F)) ) ).

% frame_inference_init
tff(fact_9819_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( fI_QUERY(P,Q,one_one_assn)
     => entails(P,Q) ) ).

% entails_solve_init(2)
tff(fact_9820_VEBT_Osize__gen_I1_J,axiom,
    ! [X11a: option4927543243414619207at_nat,X12: nat,X13a: list_VEBT_VEBT,X14a: vEBT_VEBT] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,aa_VEB5832953974749077948T_VEBT(vEBT_Node(X11a,X12,X13a),X14a)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_list_VEBT_VEBT(vEBT_size_VEBT,X13a)),aa_VEBT_VEBT_nat(vEBT_size_VEBT,X14a))),aa_nat_nat(suc,zero_zero_nat)) ).

% VEBT.size_gen(1)
tff(fact_9821_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa_VEBT_VEBT_nat(vEBT_size_VEBT,aa_o_VEBT_VEBT(vEBT_Leaf((X21)),(X222))) = zero_zero_nat ).

% VEBT.size_gen(2)
tff(fact_9822_smod__int__range,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => aa_set_int_o(member_int(aa_int_int(signed6292675348222524329lo_int(A2),B2)),set_or1266510415728281911st_int(aa_int_int(plus_plus_int(aa_int_int(uminus_uminus_int,aa_int_int(abs_abs_int,B2))),one_one_int),aa_int_int(minus_minus_int(aa_int_int(abs_abs_int,B2)),one_one_int))) ) ).

% smod_int_range
tff(fact_9823_smod__int__compares_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(signed6292675348222524329lo_int(A2),B2)) ) ) ).

% smod_int_compares(8)
tff(fact_9824_smod__int__compares_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(signed6292675348222524329lo_int(A2),B2)),zero_zero_int) ) ) ).

% smod_int_compares(7)
tff(fact_9825_smod__int__compares_I6_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(signed6292675348222524329lo_int(A2),B2)) ) ) ).

% smod_int_compares(6)
tff(fact_9826_smod__int__compares_I4_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(signed6292675348222524329lo_int(A2),B2)),zero_zero_int) ) ) ).

% smod_int_compares(4)
tff(fact_9827_smod__int__compares_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(signed6292675348222524329lo_int(A2),B2)) ) ) ).

% smod_int_compares(2)
tff(fact_9828_smod__int__compares_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(signed6292675348222524329lo_int(A2),B2)),B2) ) ) ).

% smod_int_compares(1)
tff(fact_9829_signed__modulo__int__def,axiom,
    ! [K: int,L: int] : aa_int_int(signed6292675348222524329lo_int(K),L) = aa_int_int(minus_minus_int(K),aa_int_int(times_times_int(aa_int_int(signed6714573509424544716de_int(K),L)),L)) ).

% signed_modulo_int_def
tff(fact_9830_smod__int__compares_I5_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(signed6292675348222524329lo_int(A2),B2)),aa_int_int(uminus_uminus_int,B2)) ) ) ).

% smod_int_compares(5)
tff(fact_9831_smod__int__compares_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(signed6292675348222524329lo_int(A2),B2)) ) ) ).

% smod_int_compares(3)
tff(fact_9832_uint32_Osize__eq__length,axiom,
    aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,aa_num_num(bit0,one)))))) = type_l796852477590012082l_num1(type_N8448461349408098053l_num1) ).

% uint32.size_eq_length
tff(fact_9833_len__num0,axiom,
    ! [Uu: itself_Numeral_num0] : type_l4264026598287037464l_num0(Uu) = zero_zero_nat ).

% len_num0
tff(fact_9834_len__of__finite__2__def,axiom,
    ! [Xc: itself_finite_2] : type_l31302759751748492nite_2(Xc) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% len_of_finite_2_def
tff(fact_9835_len__of__finite__3__def,axiom,
    ! [Xc: itself_finite_3] : type_l31302759751748493nite_3(Xc) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one))) ).

% len_of_finite_3_def
tff(fact_9836_min__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,M)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb)) ).

% min_Suc_Suc
tff(fact_9837_min__0R,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% min_0R
tff(fact_9838_min__0L,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),Nb) = zero_zero_nat ).

% min_0L
tff(fact_9839_min__minus_H,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)),M) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus'
tff(fact_9840_min__minus,axiom,
    ! [M: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),K) ).

% min_minus
tff(fact_9841_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),aa_nat_nat(suc,A2)) = aa_nat_nat(suc,A2) ) ) ).

% min_Suc_gt(2)
tff(fact_9842_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,A2)),B2) = aa_nat_nat(suc,A2) ) ) ).

% min_Suc_gt(1)
tff(fact_9843_rev__min__pm1,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),A2)) = A2 ).

% rev_min_pm1
tff(fact_9844_rev__min__pm,axiom,
    ! [B2: nat,A2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,B2),A2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ).

% rev_min_pm
tff(fact_9845_min__pm1,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)) = A2 ).

% min_pm1
tff(fact_9846_min__pm,axiom,
    ! [A2: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,A2),B2)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A2),B2)) = A2 ).

% min_pm
tff(fact_9847_min__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_num_nat(pred_numeral,K)),Nb)) ).

% min_numeral_Suc
tff(fact_9848_min__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),aa_num_nat(pred_numeral,K))) ).

% min_Suc_numeral
tff(fact_9849_concat__bit__assoc__sym,axiom,
    ! [M: nat,Nb: nat,K: int,L: int,R3: int] : aa_int_int(bit_concat_bit(M,aa_int_int(bit_concat_bit(Nb,K),L)),R3) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb),K),aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb),L),R3)) ).

% concat_bit_assoc_sym
tff(fact_9850_min__diff,axiom,
    ! [M: nat,I: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb)),I) ).

% min_diff
tff(fact_9851_nat__mult__min__right,axiom,
    ! [M: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),Q3)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)) ).

% nat_mult_min_right
tff(fact_9852_nat__mult__min__left,axiom,
    ! [M: nat,Nb: nat,Q3: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb)),Q3) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M),Q3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ).

% nat_mult_min_left
tff(fact_9853_take__bit__concat__bit__eq,axiom,
    ! [M: nat,Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(M),aa_int_int(bit_concat_bit(Nb,K),L)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb),K),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M),Nb)),L)) ).

% take_bit_concat_bit_eq
tff(fact_9854_mod__mod__power,axiom,
    ! [K: nat,M: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,M),Nb))) ).

% mod_mod_power
tff(fact_9855_shiftl__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh3965577149348748681tl_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% shiftl_Suc_0
tff(fact_9856_shiftr__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_Sh2154871086232339855tr_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_o_nat(zero_n2687167440665602831ol_nat,Nb = zero_zero_nat) ).

% shiftr_Suc_0
tff(fact_9857_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N: nat] :
          ( aa_set_nat_o(member_nat(N),N5)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N) )
     => aa_set_nat_o(inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kl,K)),N5) ) ).

% inj_on_diff_nat
tff(fact_9858_inj__Suc,axiom,
    ! [N5: set_nat] : aa_set_nat_o(inj_on_nat_nat(suc),N5) ).

% inj_Suc
tff(fact_9859_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_set_real_o(inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_ke,Nb)),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9860_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Ta),D2)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),D2) ) ).

% valid_eq2
tff(fact_9861_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),D2)
     => aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Ta),D2) ) ).

% valid_eq1
tff(fact_9862_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,Ta),D2)
    <=> aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_invar_vebt,Ta),D2) ) ).

% valid_eq
tff(fact_9863_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o,D2: nat] :
      ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_VEBT_valid,aa_o_VEBT_VEBT(vEBT_Leaf((Uu)),(Uv2))),D2)
    <=> ( D2 = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_9864_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xc: real,D: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( Xc != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
             => ( D = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real)
               => ( D = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
               => ( D = aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D,topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9865_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9866_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: fun_real_real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(V,K,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(V,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A2),B2)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(V,A2)),aa_real_real(V,B2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9867_DERIV__const__ratio__const,axiom,
    ! [A2: real,B2: real,F2: fun_real_real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(F2,K,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F2,B2)),aa_real_real(F2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9868_DERIV__ln,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xc),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9869_DERIV__neg__dec__right,axiom,
    ! [F2: fun_real_real,L: real,Xc: real] :
      ( has_fi5821293074295781190e_real(F2,L,topolo2177554685111907308n_real(Xc,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),H6))),aa_real_real(F2,Xc)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9870_DERIV__pos__inc__right,axiom,
    ! [F2: fun_real_real,L: real,Xc: real] :
      ( has_fi5821293074295781190e_real(F2,L,topolo2177554685111907308n_real(Xc,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,Xc)),aa_real_real(F2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),H6))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9871_DERIV__isconst__all,axiom,
    ! [F2: fun_real_real,Xc: real,Ya: real] :
      ( ! [X3: real] : has_fi5821293074295781190e_real(F2,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real))
     => ( aa_real_real(F2,Xc) = aa_real_real(F2,Ya) ) ) ).

% DERIV_isconst_all
tff(fact_9872_DERIV__mirror,axiom,
    ! [F2: fun_real_real,Ya: real,Xc: real] :
      ( has_fi5821293074295781190e_real(F2,Ya,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,Xc),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_ku(F2),aa_real_real(uminus_uminus_real,Ya),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_9873_DERIV__local__const,axiom,
    ! [F2: fun_real_real,L: real,Xc: real,D2: real] :
      ( has_fi5821293074295781190e_real(F2,L,topolo2177554685111907308n_real(Xc,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
       => ( ! [Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Y))),D2)
             => ( aa_real_real(F2,Xc) = aa_real_real(F2,Y) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9874_DERIV__pos__inc__left,axiom,
    ! [F2: fun_real_real,L: real,Xc: real] :
      ( has_fi5821293074295781190e_real(F2,L,topolo2177554685111907308n_real(Xc,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),H6))),aa_real_real(F2,Xc)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9875_DERIV__neg__dec__left,axiom,
    ! [F2: fun_real_real,L: real,Xc: real] :
      ( has_fi5821293074295781190e_real(F2,L,topolo2177554685111907308n_real(Xc,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D5)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,Xc)),aa_real_real(F2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),H6))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9876_DERIV__const__ratio__const2,axiom,
    ! [A2: real,B2: real,F2: fun_real_real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(F2,K,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F2,B2)),aa_real_real(F2,A2))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_9877_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun_real_real,L: real,Xc: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F2,L,topolo2177554685111907308n_real(Xc,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),H6)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,Xc)),aa_real_real(F2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),H6))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9878_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun_real_real,L: real,Xc: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F2,L,topolo2177554685111907308n_real(Xc,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),H6)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),H6))),aa_real_real(F2,Xc)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9879_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun_real_real,L: real,Xc: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F2,L,topolo2177554685111907308n_real(Xc,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),H6)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,Xc)),aa_real_real(F2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),H6))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9880_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun_real_real,L: real,Xc: real,S: set_real] :
      ( has_fi5821293074295781190e_real(F2,L,topolo2177554685111907308n_real(Xc,S))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
            & ! [H6: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H6)
               => ( aa_set_real_o(member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),H6)),S)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H6),D5)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xc),H6))),aa_real_real(F2,Xc)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9881_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_or1222579329274155063t_real(A2,B2))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),set_or1222579329274155063t_real(A2,B2))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(G2,X3)) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,A2)),aa_real_real(G,B2)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9882_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F2,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F2,B2)),aa_real_real(F2,A2)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9883_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F2,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y3) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F2,A2)),aa_real_real(F2,B2)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9884_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F2,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,B2)),aa_real_real(F2,A2)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9885_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F2,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,A2)),aa_real_real(F2,B2)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9886_MVT2,axiom,
    ! [A2: real,B2: real,F2: fun_real_real,F5: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2)
             => has_fi5821293074295781190e_real(F2,aa_real_real(F5,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
       => ? [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Z2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F2,B2)),aa_real_real(F2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),aa_real_real(F5,Z2)) ) ) ) ) ).

% MVT2
tff(fact_9887_DERIV__local__max,axiom,
    ! [F2: fun_real_real,L: real,Xc: real,D2: real] :
      ( has_fi5821293074295781190e_real(F2,L,topolo2177554685111907308n_real(Xc,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
       => ( ! [Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Y))),D2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F2,Y)),aa_real_real(F2,Xc)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9888_DERIV__local__min,axiom,
    ! [F2: fun_real_real,L: real,Xc: real,D2: real] :
      ( has_fi5821293074295781190e_real(F2,L,topolo2177554685111907308n_real(Xc,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
       => ( ! [Y: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Y))),D2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F2,Xc)),aa_real_real(F2,Y)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9889_DERIV__ln__divide,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xc),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9890_DERIV__pow,axiom,
    ! [Nb: nat,Xc: real,S2: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_kv,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xc,S2)) ).

% DERIV_pow
tff(fact_9891_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,M: real,Xc: real,Nb: nat] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xc,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_kw(G),Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,Xc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)))),M),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9892_has__real__derivative__powr,axiom,
    ! [Z: real,R3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Z)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_kx,R3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Z),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R3),one_one_real))),topolo2177554685111907308n_real(Z,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9893_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,M: real,Xc: real,R3: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xc,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xc))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_ky(G),R3),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xc)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R3),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),M),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9894_DERIV__log,axiom,
    ! [Xc: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(ln_ln_real,B2)),Xc)),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9895_DERIV__powr,axiom,
    ! [G: fun_real_real,M: real,Xc: real,F2: fun_real_real,R3: real] :
      ( has_fi5821293074295781190e_real(G,M,topolo2177554685111907308n_real(Xc,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xc))
       => ( has_fi5821293074295781190e_real(F2,R3,topolo2177554685111907308n_real(Xc,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_kz(G),F2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xc)),aa_real_real(F2,Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(ln_ln_real,aa_real_real(G,Xc)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,M),aa_real_real(F2,Xc))),aa_real_real(G,Xc)))),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9896_DERIV__real__sqrt,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xc))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9897_DERIV__arctan,axiom,
    ! [Xc: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xc,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9898_arsinh__real__has__field__derivative,axiom,
    ! [Xc: real,A: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xc,A)) ).

% arsinh_real_has_field_derivative
tff(fact_9899_DERIV__real__sqrt__generic,axiom,
    ! [Xc: real,D: real] :
      ( ( Xc != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
         => ( D = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xc))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),zero_zero_real)
           => ( D = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xc)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D,topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9900_arcosh__real__has__field__derivative,axiom,
    ! [Xc: real,A: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xc,A)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9901_artanh__real__has__field__derivative,axiom,
    ! [Xc: real,A: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xc,A)) ) ).

% artanh_real_has_field_derivative
tff(fact_9902_DERIV__real__root,axiom,
    ! [Nb: nat,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xc)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9903_DERIV__arccos,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),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(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9904_DERIV__arcsin,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9905_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F2: fun_real_real,Xc: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
        & ! [M4: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ? [T5: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xc))
          & ( aa_real_real(F2,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_la(Diff),Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9906_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F2: fun_real_real,Xc: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
     => ( ! [M4: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xc))
            & ( aa_real_real(F2,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_la(Diff),Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9907_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xc: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( Xc != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xc)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9908_Maclaurin,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
         => ( ! [M4: nat,T5: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),H)
                & ( aa_real_real(F2,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_lb(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9909_Maclaurin2,axiom,
    ! [H: real,Diff: fun_na6724076748692969250l_real,F2: fun_real_real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
       => ( ! [M4: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ? [T5: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T5)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H)
              & ( aa_real_real(F2,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_lb(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9910_Maclaurin__minus,axiom,
    ! [H: real,Nb: nat,Diff: fun_na6724076748692969250l_real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),zero_zero_real)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
         => ( ! [M4: nat,T5: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),zero_zero_real)
                & ( aa_real_real(F2,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_lb(H),Diff),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9911_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F2: fun_real_real,Nb: nat,Xc: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( Xc != zero_zero_real )
         => ( ! [M4: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
           => ? [T5: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T5))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xc))
                & ( aa_real_real(F2,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_la(Diff),Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9912_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F2: fun_real_real,Nb: nat,Xc: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
     => ( ! [M4: nat,T5: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xc)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
       => ? [T5: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T5)),aa_real_real(abs_abs_real,Xc))
            & ( aa_real_real(F2,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_la(Diff),Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xc),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9913_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F2: fun_real_real,A2: real,B2: real,C2: real,Xc: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
       => ( ! [M4: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),B2) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),C2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Xc)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),B2)
                 => ( ( Xc != C2 )
                   => ? [T5: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),C2),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),T5)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),C2) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),T5)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),Xc) ) )
                        & ( aa_real_real(F2,Xc) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_lc(Diff),C2),Xc),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),C2)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9914_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F2: fun_real_real,A2: real,B2: real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
       => ( ! [M4: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),B2) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),C2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),B2)
             => ? [T5: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C2),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),B2)
                  & ( aa_real_real(F2,B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ld(Diff),B2),C2),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9915_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F2: fun_real_real,A2: real,B2: real,C2: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F2 )
       => ( ! [M4: nat,T5: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),T5)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),B2) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),C2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C2),B2)
             => ? [T5: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),T5)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T5),C2)
                  & ( aa_real_real(F2,A2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ld(Diff),A2),C2),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T5)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A2),C2)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_9916_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B: real] :
      ( ! [M4: nat,T5: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T5)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T5),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M4),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T5),topolo2177554685111907308n_real(T5,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K) )
       => ! [M2: nat,T6: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T6)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T6),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_lf(Nb),Diff),B),M2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M2)),T6)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lg(Diff),M2),T6),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T6),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M2))))))),topolo2177554685111907308n_real(T6,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9917_DERIV__arctan__series,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_lh,suminf_real(aa_real_fun_nat_real(aTP_Lamm_li,Xc)),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9918_DERIV__power__series_H,axiom,
    ! [R: real,F2: fun_nat_real,X0: real] :
      ( ! [X3: real] :
          ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_lj(F2),X3)) )
     => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
         => has_fi5821293074295781190e_real(aTP_Lamm_ll(F2),suminf_real(aa_real_fun_nat_real(aTP_Lamm_lj(F2),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9919_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,Xc: real,Ya: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( aa_set_real_o(member_real(Xc),set_or1633881224788618240n_real(A2,B2))
       => ( aa_set_real_o(member_real(Ya),set_or1633881224788618240n_real(A2,B2))
         => ( ! [X3: real] :
                ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(A2,B2))
               => has_fi5821293074295781190e_real(F2,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) )
           => ( aa_real_real(F2,Xc) = aa_real_real(F2,Ya) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9920_DERIV__series_H,axiom,
    ! [F2: fun_re4238438383274954274t_real,F5: fun_re4238438383274954274t_real,X0: real,A2: real,B2: real,L5: fun_nat_real] :
      ( ! [N: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_lm(F2),N),aa_nat_real(aa_real_fun_nat_real(F5,X0),N),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(A2,B2))
           => summable_real(aa_real_fun_nat_real(F2,X3)) )
       => ( aa_set_real_o(member_real(X0),set_or1633881224788618240n_real(A2,B2))
         => ( summable_real(aa_real_fun_nat_real(F5,X0))
           => ( summable_real(L5)
             => ( ! [N: nat,X3: real,Y: real] :
                    ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(A2,B2))
                   => ( aa_set_real_o(member_real(Y),set_or1633881224788618240n_real(A2,B2))
                     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(F2,X3),N)),aa_nat_real(aa_real_fun_nat_real(F2,Y),N)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),Y)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_ln(F2),suminf_real(aa_real_fun_nat_real(F5,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9921_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) ).

% finite_greaterThanLessThan
tff(fact_9922_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,set_or5832277885323065728an_int(L,U)) ).

% finite_greaterThanLessThan_int
tff(fact_9923_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or4266950643985792945nteger(L,U)) ).

% finite_greaterThanLessThan_integer
tff(fact_9924_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or5834768355832116004an_nat(L),U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_9925_isCont__Lb__Ub,axiom,
    ! [A2: real,B2: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( ! [X3: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F2) )
       => ? [L6: real,M10: real] :
            ( ! [X4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B2) )
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_real_real(F2,X4))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F2,X4)),M10) ) )
            & ! [Y3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),Y3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),M10) )
               => ? [X3: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X3)
                    & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2)
                    & ( aa_real_real(F2,X3) = Y3 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_9926_LIM__fun__gt__zero,axiom,
    ! [F2: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F2,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [R2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X4))),R2) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F2,X4)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9927_LIM__fun__not__zero,axiom,
    ! [F2: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F2,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X4))),R2) )
               => ( aa_real_real(F2,X4) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9928_LIM__fun__less__zero,axiom,
    ! [F2: fun_real_real,L: real,C2: real] :
      ( filterlim_real_real(F2,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C2,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [R2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
            & ! [X4: real] :
                ( ( ( X4 != C2 )
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C2),X4))),R2) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,X4)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9929_isCont__real__sqrt,axiom,
    ! [Xc: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xc,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_9930_isCont__real__root,axiom,
    ! [Xc: real,Nb: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xc,top_top_set_real),aa_nat_fun_real_real(root,Nb)) ).

% isCont_real_root
tff(fact_9931_continuous__frac,axiom,
    ! [Xc: real] :
      ( ~ aa_set_real_o(member_real(Xc),ring_1_Ints_real)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xc,top_top_set_real),archim2898591450579166408c_real) ) ).

% continuous_frac
tff(fact_9932_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(plus_plus_int(L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9933_isCont__inverse__function2,axiom,
    ! [A2: real,Xc: real,B2: real,G: fun_real_real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),B2)
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
               => ( aa_real_real(G,aa_real_real(F2,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F2) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F2,Xc),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9934_isCont__ln,axiom,
    ! [Xc: real] :
      ( ( Xc != zero_zero_real )
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xc,top_top_set_real),ln_ln_real) ) ).

% isCont_ln
tff(fact_9935_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(plus_p5714425477246183910nteger(L,one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_9936_isCont__arcosh,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xc,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9937_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_lo,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),top_top_set_real)) ).

% LIM_cos_div_sin
tff(fact_9938_DERIV__inverse__function,axiom,
    ! [F2: fun_real_real,D: real,G: fun_real_real,Xc: real,A2: real,B2: real] :
      ( has_fi5821293074295781190e_real(F2,D,topolo2177554685111907308n_real(aa_real_real(G,Xc),top_top_set_real))
     => ( ( D != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Xc)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),B2)
           => ( ! [Y: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Y)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),B2)
                   => ( aa_real_real(F2,aa_real_real(G,Y)) = Y ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xc,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9939_isCont__arccos,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xc,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9940_isCont__arcsin,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xc,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9941_LIM__less__bound,axiom,
    ! [B2: real,Xc: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),Xc)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),set_or1633881224788618240n_real(B2,Xc))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F2,X3)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xc,top_top_set_real),F2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F2,Xc)) ) ) ) ).

% LIM_less_bound
tff(fact_9942_isCont__artanh,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xc,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9943_isCont__inverse__function,axiom,
    ! [D2: real,Xc: real,G: fun_real_real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),Xc))),D2)
           => ( aa_real_real(G,aa_real_real(F2,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z2),Xc))),D2)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F2) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F2,Xc),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9944_GMVT_H,axiom,
    ! [A2: real,B2: real,F2: fun_real_real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ! [Z2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Z2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),F2) ) )
       => ( ! [Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Z2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z2),B2)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z2,top_top_set_real),G) ) )
         => ( ! [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Z2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
           => ( ! [Z2: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Z2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
                   => has_fi5821293074295781190e_real(F2,aa_real_real(F5,Z2),topolo2177554685111907308n_real(Z2,top_top_set_real)) ) )
             => ? [C4: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),C4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C4),B2)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F2,B2)),aa_real_real(F2,A2))),aa_real_real(G2,C4)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B2)),aa_real_real(G,A2))),aa_real_real(F5,C4)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9945_summable__Leibniz_I3_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(A2,zero_zero_nat)),zero_zero_real)
         => ! [N8: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_lp(A2))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_lp(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)),one_one_nat))),groups6591440286371151544t_real(aTP_Lamm_lp(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_9946_summable__Leibniz_I2_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(A2,zero_zero_nat))
         => ! [N8: nat] : aa_set_real_o(member_real(suminf_real(aTP_Lamm_lp(A2))),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_lp(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8))),groups6591440286371151544t_real(aTP_Lamm_lp(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)),one_one_nat))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_9947_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9948_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_left_at_top
tff(fact_9949_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),C2)
     => filterlim_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cx,C2),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9950_monoseq__convergent,axiom,
    ! [X: fun_nat_real,B: real] :
      ( topolo6980174941875973593q_real(X)
     => ( ! [I5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X,I5))),B)
       => ~ ! [L6: real] : ~ filterlim_nat_real(X,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_9951_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_lq,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_9952_nested__sequence__unique,axiom,
    ! [F2: fun_nat_real,G: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F2,N)),aa_nat_real(F2,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N))),aa_nat_real(G,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F2,N)),aa_nat_real(G,N))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lr(F2),G),topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
           => ? [L3: real] :
                ( ! [N8: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F2,N8)),L3)
                & filterlim_nat_real(F2,topolo2815343760600316023s_real(L3),at_top_nat)
                & ! [N8: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),aa_nat_real(G,N8))
                & filterlim_nat_real(G,topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_9953_LIMSEQ__inverse__zero,axiom,
    ! [X: fun_nat_real] :
      ( ! [R2: real] :
        ? [N7: nat] :
        ! [N: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R2),aa_nat_real(X,N)) )
     => filterlim_nat_real(aTP_Lamm_ls(X),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9954_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_lt,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_9955_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_lu,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9956_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),C2)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_lv,C2),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9957_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R3: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_lw,R3),topolo2815343760600316023s_real(R3),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9958_increasing__LIMSEQ,axiom,
    ! [F2: fun_nat_real,L: real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F2,N)),aa_nat_real(F2,aa_nat_nat(suc,N)))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F2,N)),L)
       => ( ! [E2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
             => ? [N8: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F2,N8)),E2)) )
         => filterlim_nat_real(F2,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9959_LIMSEQ__realpow__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xc),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xc),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9960_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xc: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_lx(Xc),A2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9961_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C2)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,C2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_9962_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,C2)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,C2)),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_9963_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xc)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ly,Xc),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9964_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R3: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_lz,R3),topolo2815343760600316023s_real(R3),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9965_tendsto__exp__limit__sequentially,axiom,
    ! [Xc: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ma,Xc),topolo2815343760600316023s_real(aa_real_real(exp_real,Xc)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_9966_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R3: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mb,R3),topolo2815343760600316023s_real(R3),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9967_summable__Leibniz_I1_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => summable_real(aTP_Lamm_lp(A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_9968_summable,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N))),aa_nat_real(A2,N))
         => summable_real(aTP_Lamm_lp(A2)) ) ) ) ).

% summable
tff(fact_9969_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_mc(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_md(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_9970_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_me(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_md(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9971_summable__Leibniz_I4_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => filterlim_nat_real(aTP_Lamm_mf(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_lp(A2))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9972_zeroseq__arctan__series,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xc)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_hn,Xc),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9973_summable__Leibniz_H_I3_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N))),aa_nat_real(A2,N))
         => filterlim_nat_real(aTP_Lamm_mf(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_lp(A2))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9974_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N))),aa_nat_real(A2,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_lp(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),suminf_real(aTP_Lamm_lp(A2))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9975_sums__alternating__upper__lower,axiom,
    ! [A2: fun_nat_real] :
      ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N))),aa_nat_real(A2,N))
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N))
       => ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
         => ? [L3: real] :
              ( ! [N8: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_lp(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)))),L3)
              & filterlim_nat_real(aTP_Lamm_mf(A2),topolo2815343760600316023s_real(L3),at_top_nat)
              & ! [N8: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),groups6591440286371151544t_real(aTP_Lamm_lp(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N8)),one_one_nat))))
              & filterlim_nat_real(aTP_Lamm_mg(A2),topolo2815343760600316023s_real(L3),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9976_summable__Leibniz_I5_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( topolo6980174941875973593q_real(A2)
       => filterlim_nat_real(aTP_Lamm_mg(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_lp(A2))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9977_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N))),aa_nat_real(A2,N))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_lp(A2))),groups6591440286371151544t_real(aTP_Lamm_lp(A2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat)))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_9978_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun_nat_real] :
      ( filterlim_nat_real(A2,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A2,N))
       => ( ! [N: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A2,aa_nat_nat(suc,N))),aa_nat_real(A2,N))
         => filterlim_nat_real(aTP_Lamm_mg(A2),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_lp(A2))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9979_real__bounded__linear,axiom,
    ! [F2: fun_real_real] :
      ( real_V5970128139526366754l_real(F2)
    <=> ? [C5: real] :
        ! [X2: real] : aa_real_real(F2,X2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X2),C5) ) ).

% real_bounded_linear
tff(fact_9980_dist__complex__def,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_real(real_V3694042436643373181omplex(Xc),Ya) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xc),Ya)) ).

% dist_complex_def
tff(fact_9981_dist__real__def,axiom,
    ! [Xc: real,Ya: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Xc),Ya) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xc),Ya)) ).

% dist_real_def
tff(fact_9982_tendsto__exp__limit__at__right,axiom,
    ! [Xc: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_mh,Xc),topolo2815343760600316023s_real(aa_real_real(exp_real,Xc)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_9983_tendsto__arctan__at__bot,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),at_bot_real) ).

% tendsto_arctan_at_bot
tff(fact_9984_ln__at__0,axiom,
    filterlim_real_real(ln_ln_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% ln_at_0
tff(fact_9985_filterlim__tan__at__right,axiom,
    filterlim_real_real(tan_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% filterlim_tan_at_right
tff(fact_9986_exp__at__bot,axiom,
    filterlim_real_real(exp_real,topolo2815343760600316023s_real(zero_zero_real),at_bot_real) ).

% exp_at_bot
tff(fact_9987_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real(inverse_inverse_real,at_bot_real,topolo2177554685111907308n_real(zero_zero_real,set_or5984915006950818249n_real(zero_zero_real))) ).

% filterlim_inverse_at_bot_neg
tff(fact_9988_log__inj,axiom,
    ! [B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => aa_set_real_o(inj_on_real_real(aa_rea7340605660300375871l_real(log,B2)),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_9989_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_9990_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F2: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2)
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F2,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) )
     => ( filterlim_real_real(F2,topolo2815343760600316023s_real(Flim),at_bot_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F2,B2)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_9991_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F2: fun_real_real,F: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F2,at_bot_real,F)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mi(Nb),F2),at_bot_real,F) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9992_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F2: fun_real_real,F: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F2,at_bot_real,F)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mi(Nb),F2),at_top_real,F) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9993_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_9994_greaterThan__0,axiom,
    aa_nat_set_nat(set_or1210151606488870762an_nat,zero_zero_nat) = image_nat_nat(suc,top_top_set_nat) ).

% greaterThan_0
tff(fact_9995_greaterThan__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_or1210151606488870762an_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_or1210151606488870762an_nat,K),insert_nat(aa_nat_nat(suc,K),bot_bot_set_nat)) ).

% greaterThan_Suc
tff(fact_9996_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_mj,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_9997_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real(inverse_inverse_real,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% filterlim_inverse_at_top_right
tff(fact_9998_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real(inverse_inverse_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)),at_top_real) ).

% filterlim_inverse_at_right_top
tff(fact_9999_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_mk,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_10000_tendsto__exp__limit__at__top,axiom,
    ! [Xc: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ml,Xc),topolo2815343760600316023s_real(aa_real_real(exp_real,Xc)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_10001_filterlim__tan__at__left,axiom,
    filterlim_real_real(tan_real,at_top_real,topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),set_or5984915006950818249n_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% filterlim_tan_at_left
tff(fact_10002_tendsto__arctan__at__top,axiom,
    filterlim_real_real(arctan,topolo2815343760600316023s_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),at_top_real) ).

% tendsto_arctan_at_top
tff(fact_10003_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F2: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),X3)
         => ? [Y3: real] :
              ( has_fi5821293074295781190e_real(F2,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) )
     => ( filterlim_real_real(F2,topolo2815343760600316023s_real(Flim),at_top_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Flim),aa_real_real(F2,B2)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_10004_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,Xc: real,G2: fun_real_real,F2: fun_real_real,F5: fun_real_real,Ya: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
     => ( eventually_real(aTP_Lamm_mm(G2),topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G2),F5),topolo2815343760600316023s_real(Ya),topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G),F2),topolo2815343760600316023s_real(Ya),topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_10005_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_mp(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_10006_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_mq(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_10007_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_mq(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_10008_eventually__sequentially,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N9: nat] :
        ! [N6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N9),N6)
         => aa_nat_o(P,N6) ) ) ).

% eventually_sequentially
tff(fact_10009_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C2),X3)
         => aa_nat_o(P,X3) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_10010_le__sequentially,axiom,
    ! [F: filter_nat] :
      ( ord_le2510731241096832064er_nat(F,at_top_nat)
    <=> ! [N9: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N9),F) ) ).

% le_sequentially
tff(fact_10011_eventually__at__right__to__0,axiom,
    ! [P: fun_real_o,A2: real] :
      ( eventually_real(P,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
    <=> eventually_real(aa_real_fun_real_o(aTP_Lamm_mr(P),A2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_10012_eventually__at__right__real,axiom,
    ! [A2: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_ms(A2),B2),topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2))) ) ).

% eventually_at_right_real
tff(fact_10013_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A2)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_ms(B2),A2),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2))) ) ).

% eventually_at_left_real
tff(fact_10014_eventually__at__top__to__right,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,at_top_real)
    <=> eventually_real(aTP_Lamm_mt(P),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_top_to_right
tff(fact_10015_eventually__at__right__to__top,axiom,
    ! [P: fun_real_o] :
      ( eventually_real(P,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
    <=> eventually_real(aTP_Lamm_mt(P),at_top_real) ) ).

% eventually_at_right_to_top
tff(fact_10016_lhopital__at__top__at__top,axiom,
    ! [F2: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F2,at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(A2,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(A2,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F5),G2),at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F2),G),at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_10017_lhopital,axiom,
    ! [F2: fun_real_real,Xc: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F: filter_real] :
      ( filterlim_real_real(F2,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xc,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xc,top_top_set_real))
       => ( eventually_real(aTP_Lamm_mm(G),topolo2177554685111907308n_real(Xc,top_top_set_real))
         => ( eventually_real(aTP_Lamm_mm(G2),topolo2177554685111907308n_real(Xc,top_top_set_real))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(Xc,top_top_set_real))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(Xc,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G2),F5),F,topolo2177554685111907308n_real(Xc,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F2),G),F,topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_10018_lhopital__right__at__top__at__top,axiom,
    ! [F2: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F2,at_top_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F5),G2),at_top_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F2),G),at_top_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_10019_lhopital__at__top__at__bot,axiom,
    ! [F2: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F2,at_top_real,topolo2177554685111907308n_real(A2,top_top_set_real))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A2,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(A2,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(A2,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F5),G2),at_bot_real,topolo2177554685111907308n_real(A2,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F2),G),at_bot_real,topolo2177554685111907308n_real(A2,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_10020_lhopital__left__at__top__at__top,axiom,
    ! [F2: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F2,at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F5),G2),at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F2),G),at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_10021_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F2: fun_real_real,F5: fun_real_real,Xc: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_mm(G2),at_top_real)
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),at_top_real)
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G2),F5),topolo2815343760600316023s_real(Xc),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G),F2),topolo2815343760600316023s_real(Xc),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_10022_lhopital__at__top,axiom,
    ! [G: fun_real_real,Xc: real,G2: fun_real_real,F2: fun_real_real,F5: fun_real_real,Ya: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xc,top_top_set_real))
     => ( eventually_real(aTP_Lamm_mm(G2),topolo2177554685111907308n_real(Xc,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(Xc,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(Xc,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G2),F5),topolo2815343760600316023s_real(Ya),topolo2177554685111907308n_real(Xc,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G),F2),topolo2815343760600316023s_real(Ya),topolo2177554685111907308n_real(Xc,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_10023_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F5: fun_real_real,F: filter_real] :
      ( filterlim_real_real(F0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( filterlim_real_real(G0,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aTP_Lamm_mm(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_mm(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F0),F5),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G2),F5),F,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F0),G0),F,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_10024_lhopital__right,axiom,
    ! [F2: fun_real_real,Xc: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F: filter_real] :
      ( filterlim_real_real(F2,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
       => ( eventually_real(aTP_Lamm_mm(G),topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
         => ( eventually_real(aTP_Lamm_mm(G2),topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G2),F5),F,topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F2),G),F,topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_10025_lhopital__left,axiom,
    ! [F2: fun_real_real,Xc: real,G: fun_real_real,G2: fun_real_real,F5: fun_real_real,F: filter_real] :
      ( filterlim_real_real(F2,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
       => ( eventually_real(aTP_Lamm_mm(G),topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
         => ( eventually_real(aTP_Lamm_mm(G2),topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G2),F5),F,topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F2),G),F,topolo2177554685111907308n_real(Xc,set_or5984915006950818249n_real(Xc))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_10026_lhopital__right__at__top__at__bot,axiom,
    ! [F2: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F2,at_top_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F5),G2),at_bot_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F2),G),at_bot_real,topolo2177554685111907308n_real(A2,set_or5849166863359141190n_real(A2))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_10027_lhopital__left__at__top__at__bot,axiom,
    ! [F2: fun_real_real,A2: real,G: fun_real_real,F5: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F2,at_top_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F5),G2),at_bot_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(F2),G),at_bot_real,topolo2177554685111907308n_real(A2,set_or5984915006950818249n_real(A2))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_10028_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F2: fun_real_real,F5: fun_real_real,Xc: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_mm(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G2),F5),topolo2815343760600316023s_real(Xc),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G),F2),topolo2815343760600316023s_real(Xc),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_10029_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,Xc: real,G2: fun_real_real,F2: fun_real_real,F5: fun_real_real,Ya: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
     => ( eventually_real(aTP_Lamm_mm(G2),topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(F2),F5),topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_mn(G),G2),topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G2),F5),topolo2815343760600316023s_real(Ya),topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(G),F2),topolo2815343760600316023s_real(Ya),topolo2177554685111907308n_real(Xc,set_or5849166863359141190n_real(Xc))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_10030_Bseq__realpow,axiom,
    ! [Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xc)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,Xc),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_10031_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_set_nat_o(finite_finite_nat,aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) ).

% finite_greaterThanAtMost
tff(fact_10032_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K: nat,B2: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B2) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_10033_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K: nat,B2: nat] :
      ( aa_nat_o(P,K)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B2) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_10034_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B2: nat] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
     => ( ! [Y: nat] :
            ( aa_nat_o(P,Y)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),B2) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_10035_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,L)),U) = aa_nat_set_nat(set_or6659071591806873216st_nat(L),U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_10036_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa_set_int_o(finite_finite_int,set_or6656581121297822940st_int(L,U)) ).

% finite_greaterThanAtMost_int
tff(fact_10037_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa_set5643387332196575111eger_o(finite6017078050557962740nteger,set_or2715278749043346189nteger(L,U)) ).

% finite_greaterThanAtMost_integer
tff(fact_10038_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(plus_plus_int(L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_10039_decseq__bounded,axiom,
    ! [X: fun_nat_real,B: real] :
      ( order_9091379641038594480t_real(X)
     => ( ! [I5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_nat_real(X,I5))
       => bfun_nat_real(X,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_10040_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or189985376899183464nteger(plus_p5714425477246183910nteger(L,one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_10041_decseq__convergent,axiom,
    ! [X: fun_nat_real,B: real] :
      ( order_9091379641038594480t_real(X)
     => ( ! [I5: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),aa_nat_real(X,I5))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I6: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_nat_real(X,I6)) ) ) ) ).

% decseq_convergent
tff(fact_10042_GMVT,axiom,
    ! [A2: real,B2: real,F2: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ! [X3: real] :
            ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X3)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F2) )
       => ( ! [X3: real] :
              ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),X3)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B2) )
             => differ6690327859849518006l_real(F2,topolo2177554685111907308n_real(X3,top_top_set_real)) )
         => ( ! [X3: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),X3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2) )
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),G) )
           => ( ! [X3: real] :
                  ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),X3)
                    & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B2) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X3,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C4: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C4,top_top_set_real))
                  & has_fi5821293074295781190e_real(F2,F_c,topolo2177554685111907308n_real(C4,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,A2),C4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C4),B2)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F2,B2)),aa_real_real(F2,A2))),G_c) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B2)),aa_real_real(G,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_10043_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_10044_real__differentiable__def,axiom,
    ! [F2: fun_real_real,Xc: real,S2: set_real] :
      ( differ6690327859849518006l_real(F2,topolo2177554685111907308n_real(Xc,S2))
    <=> ? [D6: real] : has_fi5821293074295781190e_real(F2,D6,topolo2177554685111907308n_real(Xc,S2)) ) ).

% real_differentiable_def
tff(fact_10045_real__differentiableE,axiom,
    ! [F2: fun_real_real,Xc: real,S2: set_real] :
      ( differ6690327859849518006l_real(F2,topolo2177554685111907308n_real(Xc,S2))
     => ~ ! [Df: real] : ~ has_fi5821293074295781190e_real(F2,Df,topolo2177554685111907308n_real(Xc,S2)) ) ).

% real_differentiableE
tff(fact_10046_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = aa_nat_set_nat(set_or1210151606488870762an_nat,K) ).

% atLeast_Suc_greaterThan
tff(fact_10047_atLeast__Suc,axiom,
    ! [K: nat] : aa_nat_set_nat(set_ord_atLeast_nat,aa_nat_nat(suc,K)) = minus_minus_set_nat(aa_nat_set_nat(set_ord_atLeast_nat,K),insert_nat(K,bot_bot_set_nat)) ).

% atLeast_Suc
tff(fact_10048_MVT,axiom,
    ! [A2: real,B2: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F2)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B2)
               => differ6690327859849518006l_real(F2,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ? [L3: real,Z2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Z2)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
              & has_fi5821293074295781190e_real(F2,L3,topolo2177554685111907308n_real(Z2,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F2,B2)),aa_real_real(F2,A2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)),L3) ) ) ) ) ) ).

% MVT
tff(fact_10049_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),aa_nat_nat(suc,I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J2))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_10050_continuous__on__arcosh_H,axiom,
    ! [A: set_real,F2: fun_real_real] :
      ( topolo5044208981011980120l_real(A,F2)
     => ( ! [X3: real] :
            ( aa_set_real_o(member_real(X3),A)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F2,X3)) )
       => topolo5044208981011980120l_real(A,aTP_Lamm_mv(F2)) ) ) ).

% continuous_on_arcosh'
tff(fact_10051_continuous__image__closed__interval,axiom,
    ! [A2: real,B2: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F2)
       => ? [C4: real,D5: real] :
            ( ( image_real_real(F2,set_or1222579329274155063t_real(A2,B2)) = set_or1222579329274155063t_real(C4,D5) )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C4),D5) ) ) ) ).

% continuous_image_closed_interval
tff(fact_10052_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I)),J2)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J2)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J2))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_10053_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),J2)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J2)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J2))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_10054_Rolle__deriv,axiom,
    ! [A2: real,B2: real,F2: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ( aa_real_real(F2,A2) = aa_real_real(F2,B2) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F2)
         => ( ! [X3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),X3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B2)
                 => has_de1759254742604945161l_real(F2,aa_rea7340605660300375871l_real(F5,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
                & ! [X4: real] : aa_real_real(aa_rea7340605660300375871l_real(F5,Z2),X4) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_10055_mvt,axiom,
    ! [A2: real,B2: real,F2: fun_real_real,F5: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F2)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B2)
               => has_de1759254742604945161l_real(F2,aa_rea7340605660300375871l_real(F5,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ~ ! [Xi3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Xi3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xi3),B2)
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F2,B2)),aa_real_real(F2,A2)) != aa_real_real(aa_rea7340605660300375871l_real(F5,Xi3),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A2)) ) ) ) ) ) ) ).

% mvt
tff(fact_10056_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B2: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B2)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F2,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y3) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,A2)),aa_real_real(F2,B2)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_10057_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B2: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),X3)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B2)
             => ? [Y3: real] :
                  ( has_fi5821293074295781190e_real(F2,Y3,topolo2177554685111907308n_real(X3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),zero_zero_real) ) ) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F2,B2)),aa_real_real(F2,A2)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_10058_DERIV__isconst__end,axiom,
    ! [A2: real,B2: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F2)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B2)
               => has_fi5821293074295781190e_real(F2,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ( aa_real_real(F2,B2) = aa_real_real(F2,A2) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_10059_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F2: fun_real_real,Xc: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F2)
       => ( ! [X3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),X3)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B2)
               => has_fi5821293074295781190e_real(F2,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A2),Xc)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xc),B2)
             => ( aa_real_real(F2,Xc) = aa_real_real(F2,A2) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_10060_Rolle,axiom,
    ! [A2: real,B2: real,F2: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B2)
     => ( ( aa_real_real(F2,A2) = aa_real_real(F2,B2) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A2,B2),F2)
         => ( ! [X3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),X3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),B2)
                 => differ6690327859849518006l_real(F2,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
           => ? [Z2: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A2),Z2)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),B2)
                & has_fi5821293074295781190e_real(F2,zero_zero_real,topolo2177554685111907308n_real(Z2,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_10061_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J2: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J2),I))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J2))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_10062_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,code_int_of_integer(uminus1351360451143612070nteger(K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_mw,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_10063_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Z)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_10064_minus__integer_Orep__eq,axiom,
    ! [Xc: code_integer,Xaa: code_integer] : code_int_of_integer(minus_8373710615458151222nteger(Xc,Xaa)) = aa_int_int(minus_minus_int(code_int_of_integer(Xc)),code_int_of_integer(Xaa)) ).

% minus_integer.rep_eq
tff(fact_10065_complex__Re__numeral,axiom,
    ! [V: num] : aa_complex_real(re,numera6690914467698888265omplex(V)) = aa_num_real(numeral_numeral_real,V) ).

% complex_Re_numeral
tff(fact_10066_divide__integer_Orep__eq,axiom,
    ! [Xc: code_integer,Xaa: code_integer] : code_int_of_integer(divide6298287555418463151nteger(Xc,Xaa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,code_int_of_integer(Xc)),code_int_of_integer(Xaa)) ).

% divide_integer.rep_eq
tff(fact_10067_Re__divide__of__nat,axiom,
    ! [Z: complex,Nb: nat] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% Re_divide_of_nat
tff(fact_10068_Re__divide__of__real,axiom,
    ! [Z: complex,R3: real] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),real_V4546457046886955230omplex(R3))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),R3) ).

% Re_divide_of_real
tff(fact_10069_Re__sgn,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).

% Re_sgn
tff(fact_10070_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_num_real(numeral_numeral_real,W)) ).

% Re_divide_numeral
tff(fact_10071_cos__Arg__i__mult__zero,axiom,
    ! [Ya: complex] :
      ( ( Ya != zero_zero_complex )
     => ( ( aa_complex_real(re,Ya) = zero_zero_real )
       => ( aa_real_real(cos_real,aa_complex_real(arg,Ya)) = zero_zero_real ) ) ) ).

% cos_Arg_i_mult_zero
tff(fact_10072_Re__csqrt,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z))) ).

% Re_csqrt
tff(fact_10073_complex__Re__le__cmod,axiom,
    ! [Xc: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xc)),aa_complex_real(real_V1022390504157884413omplex,Xc)) ).

% complex_Re_le_cmod
tff(fact_10074_abs__Re__le__cmod,axiom,
    ! [Xc: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,Xc))),aa_complex_real(real_V1022390504157884413omplex,Xc)) ).

% abs_Re_le_cmod
tff(fact_10075_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_iff
tff(fact_10076_less__integer_Orep__eq,axiom,
    ! [Xc: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xc),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(Xc)),code_int_of_integer(Xaa)) ) ).

% less_integer.rep_eq
tff(fact_10077_int__of__integer__less__iff,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(Xc)),code_int_of_integer(Ya))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xc),Ya) ) ).

% int_of_integer_less_iff
tff(fact_10078_minus__complex_Osimps_I1_J,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_real(re,aa_complex_complex(minus_minus_complex(Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,Xc)),aa_complex_real(re,Ya)) ).

% minus_complex.simps(1)
tff(fact_10079_zero__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(1)
tff(fact_10080_imaginary__unit_Osimps_I1_J,axiom,
    aa_complex_real(re,imaginary_unit) = zero_zero_real ).

% imaginary_unit.simps(1)
tff(fact_10081_plus__complex_Osimps_I1_J,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_real(re,aa_complex_complex(plus_plus_complex(Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,Xc)),aa_complex_real(re,Ya)) ).

% plus_complex.simps(1)
tff(fact_10082_scaleR__complex_Osimps_I1_J,axiom,
    ! [R3: real,Xc: complex] : aa_complex_real(re,aa_complex_complex(real_V2046097035970521341omplex(R3),Xc)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_complex_real(re,Xc)) ).

% scaleR_complex.simps(1)
tff(fact_10083_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),zero_zero_real)
    <=> ( aa_complex_real(re,Z) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_10084_bin__last__integer_Orep__eq,axiom,
    ! [Xc: code_integer] :
      ( aa_Code_integer_o(bits_b8758750999018896077nteger,Xc)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Xc)) ) ).

% bin_last_integer.rep_eq
tff(fact_10085_bin__rest__integer_Orep__eq,axiom,
    ! [Xc: code_integer] : code_int_of_integer(bits_b2549910563261871055nteger(Xc)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,code_int_of_integer(Xc)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% bin_rest_integer.rep_eq
tff(fact_10086_cos__n__Re__cis__pow__n,axiom,
    ! [Nb: nat,A2: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A2)) = aa_complex_real(re,aa_nat_complex(power_power_complex(cis(A2)),Nb)) ).

% cos_n_Re_cis_pow_n
tff(fact_10087_Bit__integer_Orep__eq,axiom,
    ! [Xc: code_integer,Xaa: $o] : code_int_of_integer(bits_Bit_integer(Xc,(Xaa))) = aa_int_int(plus_plus_int(aa_o_int(zero_n2684676970156552555ol_int,(Xaa))),aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Xc))) ).

% Bit_integer.rep_eq
tff(fact_10088_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = produc1086072967326762835nteger(divide6298287555418463151nteger(K,L),modulo364778990260209775nteger(K,L)) ).

% divmod_integer_def
tff(fact_10089_num__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_num(code_num_of_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_mx,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_10090_csqrt_Ocode,axiom,
    ! [Z: complex] :
      aa_complex_complex(csqrt,Z) = complex2(aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
            $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_10091_complex__Im__fact,axiom,
    ! [Nb: nat] : aa_complex_real(im,aa_nat_complex(semiri5044797733671781792omplex,Nb)) = zero_zero_real ).

% complex_Im_fact
tff(fact_10092_complex__Im__of__int,axiom,
    ! [Z: int] : aa_complex_real(im,ring_17405671764205052669omplex(Z)) = zero_zero_real ).

% complex_Im_of_int
tff(fact_10093_Im__complex__of__real,axiom,
    ! [Z: real] : aa_complex_real(im,real_V4546457046886955230omplex(Z)) = zero_zero_real ).

% Im_complex_of_real
tff(fact_10094_Im__power__real,axiom,
    ! [Xc: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xc) = zero_zero_real )
     => ( aa_complex_real(im,aa_nat_complex(power_power_complex(Xc),Nb)) = zero_zero_real ) ) ).

% Im_power_real
tff(fact_10095_complex__Im__numeral,axiom,
    ! [V: num] : aa_complex_real(im,numera6690914467698888265omplex(V)) = zero_zero_real ).

% complex_Im_numeral
tff(fact_10096_complex__Im__of__nat,axiom,
    ! [Nb: nat] : aa_complex_real(im,aa_nat_complex(semiri8010041392384452111omplex,Nb)) = zero_zero_real ).

% complex_Im_of_nat
tff(fact_10097_Im__divide__of__real,axiom,
    ! [Z: complex,R3: real] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),real_V4546457046886955230omplex(R3))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),R3) ).

% Im_divide_of_real
tff(fact_10098_Im__sgn,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(sgn_sgn_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z)) ).

% Im_sgn
tff(fact_10099_Re__power__real,axiom,
    ! [Xc: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xc) = zero_zero_real )
     => ( aa_complex_real(re,aa_nat_complex(power_power_complex(Xc),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xc)),Nb) ) ) ).

% Re_power_real
tff(fact_10100_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_num_real(numeral_numeral_real,W)) ).

% Im_divide_numeral
tff(fact_10101_Im__divide__of__nat,axiom,
    ! [Z: complex,Nb: nat] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% Im_divide_of_nat
tff(fact_10102_csqrt__of__real__nonneg,axiom,
    ! [Xc: complex] :
      ( ( aa_complex_real(im,Xc) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xc))
       => ( aa_complex_complex(csqrt,Xc) = real_V4546457046886955230omplex(aa_real_real(sqrt,aa_complex_real(re,Xc))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_10103_csqrt__minus,axiom,
    ! [Xc: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,Xc)),zero_zero_real)
        | ( ( aa_complex_real(im,Xc) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xc)) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,Xc)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,Xc)) ) ) ).

% csqrt_minus
tff(fact_10104_csqrt__of__real__nonpos,axiom,
    ! [Xc: complex] :
      ( ( aa_complex_real(im,Xc) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xc)),zero_zero_real)
       => ( aa_complex_complex(csqrt,Xc) = aa_complex_complex(times_times_complex(imaginary_unit),real_V4546457046886955230omplex(aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,Xc))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_10105_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( aa_set_complex_o(member_complex(Z),ring_1_Ints_complex)
    <=> ( ( aa_complex_real(im,Z) = zero_zero_real )
        & ? [I2: int] : aa_complex_real(re,Z) = aa_int_real(ring_1_of_int_real,I2) ) ) ).

% complex_is_Int_iff
tff(fact_10106_zero__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,zero_zero_complex) = zero_zero_real ).

% zero_complex.simps(2)
tff(fact_10107_one__complex_Osimps_I2_J,axiom,
    aa_complex_real(im,one_one_complex) = zero_zero_real ).

% one_complex.simps(2)
tff(fact_10108_plus__complex_Osimps_I2_J,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_real(im,aa_complex_complex(plus_plus_complex(Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,Xc)),aa_complex_real(im,Ya)) ).

% plus_complex.simps(2)
tff(fact_10109_scaleR__complex_Osimps_I2_J,axiom,
    ! [R3: real,Xc: complex] : aa_complex_real(im,aa_complex_complex(real_V2046097035970521341omplex(R3),Xc)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_complex_real(im,Xc)) ).

% scaleR_complex.simps(2)
tff(fact_10110_minus__complex_Osimps_I2_J,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_real(im,aa_complex_complex(minus_minus_complex(Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,Xc)),aa_complex_real(im,Ya)) ).

% minus_complex.simps(2)
tff(fact_10111_abs__Im__le__cmod,axiom,
    ! [Xc: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,Xc))),aa_complex_real(real_V1022390504157884413omplex,Xc)) ).

% abs_Im_le_cmod
tff(fact_10112_times__complex_Osimps_I2_J,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xc)),aa_complex_real(im,Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xc)),aa_complex_real(re,Ya))) ).

% times_complex.simps(2)
tff(fact_10113_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( aa_real_real(abs_abs_real,aa_complex_real(re,Z)) = aa_complex_real(real_V1022390504157884413omplex,Z) )
     => ( aa_complex_real(im,Z) = zero_zero_real ) ) ).

% Im_eq_0
tff(fact_10114_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(re,Z) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(abs_abs_real,aa_complex_real(im,Z)) ) ) ).

% cmod_eq_Im
tff(fact_10115_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( aa_complex_real(im,Z) = zero_zero_real )
     => ( aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(abs_abs_real,aa_complex_real(re,Z)) ) ) ).

% cmod_eq_Re
tff(fact_10116_cmod__Im__le__iff,axiom,
    ! [Xc: complex,Ya: complex] :
      ( ( aa_complex_real(re,Xc) = aa_complex_real(re,Ya) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),aa_complex_real(real_V1022390504157884413omplex,Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,Xc))),aa_real_real(abs_abs_real,aa_complex_real(im,Ya))) ) ) ).

% cmod_Im_le_iff
tff(fact_10117_cmod__Re__le__iff,axiom,
    ! [Xc: complex,Ya: complex] :
      ( ( aa_complex_real(im,Xc) = aa_complex_real(im,Ya) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xc)),aa_complex_real(real_V1022390504157884413omplex,Ya))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,Xc))),aa_real_real(abs_abs_real,aa_complex_real(re,Ya))) ) ) ).

% cmod_Re_le_iff
tff(fact_10118_times__complex_Osimps_I1_J,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xc)),aa_complex_real(re,Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xc)),aa_complex_real(im,Ya))) ).

% times_complex.simps(1)
tff(fact_10119_plus__complex_Ocode,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_complex(plus_plus_complex(Xc),Ya) = complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,Xc)),aa_complex_real(re,Ya)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,Xc)),aa_complex_real(im,Ya))) ).

% plus_complex.code
tff(fact_10120_scaleR__complex_Ocode,axiom,
    ! [R3: real,Xc: complex] : aa_complex_complex(real_V2046097035970521341omplex(R3),Xc) = complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_complex_real(re,Xc)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_complex_real(im,Xc))) ).

% scaleR_complex.code
tff(fact_10121_minus__complex_Ocode,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_complex(minus_minus_complex(Xc),Ya) = complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,Xc)),aa_complex_real(re,Ya)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,Xc)),aa_complex_real(im,Ya))) ).

% minus_complex.code
tff(fact_10122_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Z)))
      | ( ( aa_complex_real(re,aa_complex_complex(csqrt,Z)) = zero_zero_real )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(csqrt,Z))) ) ) ).

% csqrt_principal
tff(fact_10123_cmod__le,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))) ).

% cmod_le
tff(fact_10124_sin__n__Im__cis__pow__n,axiom,
    ! [Nb: nat,A2: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A2)) = aa_complex_real(im,aa_nat_complex(power_power_complex(cis(A2)),Nb)) ).

% sin_n_Im_cis_pow_n
tff(fact_10125_Re__exp,axiom,
    ! [Z: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(cos_real,aa_complex_real(im,Z))) ).

% Re_exp
tff(fact_10126_Im__exp,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Z))),aa_real_real(sin_real,aa_complex_real(im,Z))) ).

% Im_exp
tff(fact_10127_times__complex_Ocode,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_complex(times_times_complex(Xc),Ya) = complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xc)),aa_complex_real(re,Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xc)),aa_complex_real(im,Ya))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xc)),aa_complex_real(im,Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xc)),aa_complex_real(re,Ya)))) ).

% times_complex.code
tff(fact_10128_cmod__power2,axiom,
    ! [Z: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_10129_Im__power2,axiom,
    ! [Xc: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Xc))),aa_complex_real(im,Xc)) ).

% Im_power2
tff(fact_10130_Re__power2,axiom,
    ! [Xc: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(Xc),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_10131_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_10132_norm__complex__def,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,Z) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_10133_inverse__complex_Osimps_I1_J,axiom,
    ! [Xc: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xc)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_10134_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% complex_neq_0
tff(fact_10135_Re__divide,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xc)),aa_complex_real(re,Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xc)),aa_complex_real(im,Ya)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_10136_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa_nat_complex(power_power_complex(W),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Z )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,W))
          | ( ( aa_complex_real(re,W) = zero_zero_real )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,W)) ) )
       => ( aa_complex_complex(csqrt,Z) = W ) ) ) ).

% csqrt_unique
tff(fact_10137_csqrt__square,axiom,
    ! [B2: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B2))
        | ( ( aa_complex_real(re,B2) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B2)) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B2 ) ) ).

% csqrt_square
tff(fact_10138_inverse__complex_Osimps_I2_J,axiom,
    ! [Xc: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,Xc)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_10139_Im__divide,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Xc),Ya)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xc)),aa_complex_real(re,Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xc)),aa_complex_real(im,Ya)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_10140_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Z))),aa_real_real(abs_abs_real,aa_complex_real(im,Z)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_complex_real(real_V1022390504157884413omplex,Z))) ).

% complex_abs_le_norm
tff(fact_10141_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(real_V1022390504157884413omplex,Z))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_10142_inverse__complex_Ocode,axiom,
    ! [Xc: complex] : aa_complex_complex(invers8013647133539491842omplex,Xc) = complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xc)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xc))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xc)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_10143_Complex__divide,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_complex(divide1717551699836669952omplex(Xc),Ya) = complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xc)),aa_complex_real(re,Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xc)),aa_complex_real(im,Ya)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xc)),aa_complex_real(re,Ya))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xc)),aa_complex_real(im,Ya)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Ya)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_10144_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      aa_complex_real(im,aa_complex_complex(csqrt,Z)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(aa_complex_real(im,Z) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Z)))),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_complex_real(re,Z))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_10145_Im__Reals__divide,axiom,
    ! [R3: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R3),real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R3),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_complex_real(re,R3))),aa_complex_real(im,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_10146_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      aa_Code_integer_nat(code_nat_of_integer,K) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_my,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_10147_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : aa_Code_integer_nat(code_nat_of_integer,numera6620942414471956472nteger(K)) = aa_num_nat(numeral_numeral_nat,K) ).

% nat_of_integer_code_post(3)
tff(fact_10148_nat__of__integer__numeral,axiom,
    ! [Nb: num] : aa_Code_integer_nat(code_nat_of_integer,numera6620942414471956472nteger(Nb)) = aa_num_nat(numeral_numeral_nat,Nb) ).

% nat_of_integer_numeral
tff(fact_10149_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,K),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_nat(code_nat_of_integer,K) = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
tff(fact_10150_Re__divide__Reals,axiom,
    ! [R3: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R3),real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Z),R3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Z)),aa_complex_real(re,R3)) ) ) ).

% Re_divide_Reals
tff(fact_10151_Im__divide__Reals,axiom,
    ! [R3: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R3),real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Z),R3)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Z)),aa_complex_real(re,R3)) ) ) ).

% Im_divide_Reals
tff(fact_10152_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( aa_set_complex_o(member_complex(Z),real_V2521375963428798218omplex)
    <=> ( aa_complex_real(im,Z) = zero_zero_real ) ) ).

% complex_is_Real_iff
tff(fact_10153_nat__of__integer__code__post_I1_J,axiom,
    aa_Code_integer_nat(code_nat_of_integer,zero_z3403309356797280102nteger) = zero_zero_nat ).

% nat_of_integer_code_post(1)
tff(fact_10154_Complex__in__Reals,axiom,
    ! [Xc: real] : aa_set_complex_o(member_complex(complex2(Xc,zero_zero_real)),real_V2521375963428798218omplex) ).

% Complex_in_Reals
tff(fact_10155_nat__of__integer__less__iff,axiom,
    ! [Xc: code_integer,Ya: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xc)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Ya)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_Code_integer_nat(code_nat_of_integer,Xc)),aa_Code_integer_nat(code_nat_of_integer,Ya))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xc),Ya) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_10156_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),U)
     => ( set_or8404916559141939852nteger(zero_z3403309356797280102nteger,U) = image_1215581382706833972nteger(semiri4939895301339042750nteger,aa_nat_set_nat(set_ord_lessThan_nat,aa_Code_integer_nat(code_nat_of_integer,U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_10157_Re__Reals__divide,axiom,
    ! [R3: complex,Z: complex] :
      ( aa_set_complex_o(member_complex(R3),real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R3),Z)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,R3)),aa_complex_real(re,Z))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_10158_complex__diff__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(minus_minus_complex(Z),aa_complex_complex(cnj,Z)) = aa_complex_complex(times_times_complex(real_V4546457046886955230omplex(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(im,Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_10159_complex__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) = real_V4546457046886955230omplex(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_10160_complex__cnj__divide,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_complex(cnj,aa_complex_complex(divide1717551699836669952omplex(Xc),Ya)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cnj,Xc)),aa_complex_complex(cnj,Ya)) ).

% complex_cnj_divide
tff(fact_10161_complex__cnj__diff,axiom,
    ! [Xc: complex,Ya: complex] : aa_complex_complex(cnj,aa_complex_complex(minus_minus_complex(Xc),Ya)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(cnj,Xc)),aa_complex_complex(cnj,Ya)) ).

% complex_cnj_diff
tff(fact_10162_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_10163_Re__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) = zero_zero_real )
    <=> ( aa_complex_real(re,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2))) = zero_zero_real ) ) ).

% Re_complex_div_eq_0
tff(fact_10164_Im__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A2),B2)) = zero_zero_real )
    <=> ( aa_complex_real(im,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2))) = zero_zero_real ) ) ).

% Im_complex_div_eq_0
tff(fact_10165_Re__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A2),B2))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2)))),zero_zero_real) ) ).

% Re_complex_div_lt_0
tff(fact_10166_Re__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A2),B2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2)))) ) ).

% Re_complex_div_gt_0
tff(fact_10167_Re__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A2),B2))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2)))),zero_zero_real) ) ).

% Re_complex_div_le_0
tff(fact_10168_Re__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A2),B2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2)))) ) ).

% Re_complex_div_ge_0
tff(fact_10169_Im__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A2),B2))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2)))),zero_zero_real) ) ).

% Im_complex_div_lt_0
tff(fact_10170_Im__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A2),B2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2)))) ) ).

% Im_complex_div_gt_0
tff(fact_10171_Im__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A2),B2))),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2)))),zero_zero_real) ) ).

% Im_complex_div_le_0
tff(fact_10172_Im__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A2),B2)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2)))) ) ).

% Im_complex_div_ge_0
tff(fact_10173_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_10174_complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A2),B2)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2)))) )
      & ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A2),B2)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2)))) ) ) ).

% complex_div_gt_0
tff(fact_10175_complex__norm__square,axiom,
    ! [Z: complex] : real_V4546457046886955230omplex(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Z)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,Z)) ).

% complex_norm_square
tff(fact_10176_complex__add__cnj,axiom,
    ! [Z: complex] : aa_complex_complex(plus_plus_complex(Z),aa_complex_complex(cnj,Z)) = real_V4546457046886955230omplex(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Z))) ).

% complex_add_cnj
tff(fact_10177_complex__div__cnj,axiom,
    ! [A2: complex,B2: complex] : aa_complex_complex(divide1717551699836669952omplex(A2),B2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A2),aa_complex_complex(cnj,B2))),real_V4546457046886955230omplex(aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_10178_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Z)),W)) = real_V4546457046886955230omplex(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,aa_complex_complex(times_times_complex(Z),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_10179_card__lessThan,axiom,
    ! [U: nat] : finite_card_nat(aa_nat_set_nat(set_ord_lessThan_nat,U)) = U ).

% card_lessThan
tff(fact_10180_card__Collect__less__nat,axiom,
    ! [Nb: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ft,Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_10181_card__atMost,axiom,
    ! [U: nat] : finite_card_nat(aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_10182_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_10183_card__Collect__le__nat,axiom,
    ! [Nb: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fs,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_10184_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_10185_card__UNIV__bool,axiom,
    finite_card_o(top_top_set_o) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% card_UNIV_bool
tff(fact_10186_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_10187_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or4662586982721622107an_int(L,U)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(U),L)) ).

% card_atLeastLessThan_int
tff(fact_10188_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_10189_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or6656581121297822940st_int(L,U)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(U),L)) ).

% card_greaterThanAtMost_int
tff(fact_10190_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(plus_plus_int(aa_int_int(minus_minus_int(U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_10191_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(minus_minus_int(U),aa_int_int(plus_plus_int(L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_10192_subset__card__intvl__is__intvl,axiom,
    ! [A: set_nat,K: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(A),aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A))))
     => ( A = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_10193_card__less,axiom,
    ! [M3: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M3)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_mz(M3),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_10194_card__less__Suc,axiom,
    ! [M3: set_nat,I: nat] :
      ( aa_set_nat_o(member_nat(zero_zero_nat),M3)
     => ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_na(M3),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_mz(M3),I))) ) ) ).

% card_less_Suc
tff(fact_10195_card__less__Suc2,axiom,
    ! [M3: set_nat,I: nat] :
      ( ~ aa_set_nat_o(member_nat(zero_zero_nat),M3)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_na(M3),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_mz(M3),I))) ) ) ).

% card_less_Suc2
tff(fact_10196_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_card_int(set_or4662586982721622107an_int(zero_zero_int,U)) = aa_int_nat(nat2,U) ).

% card_atLeastZeroLessThan_int
tff(fact_10197_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set_nat,Nb: nat] :
      ( aa_set_nat_o(ord_less_eq_set_nat(N5),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(N5)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_10198_card__le__Suc__Max,axiom,
    ! [S: set_nat] :
      ( aa_set_nat_o(finite_finite_nat,S)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(S)),aa_nat_nat(suc,lattic8265883725875713057ax_nat(S))) ) ).

% card_le_Suc_Max
tff(fact_10199_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cn),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),finite_card_nat(S)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_cn),S)) ).

% card_sum_le_nat_sum
tff(fact_10200_card__nth__roots,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( finite_card_complex(collect_complex(aa_nat_fun_complex_o(aTP_Lamm_jt(C2),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_10201_card__roots__unity__eq,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( finite_card_complex(collect_complex(aTP_Lamm_fr(Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_10202_card__num0,axiom,
    finite6454714172617411596l_num0(top_to3689904424835650196l_num0) = zero_zero_nat ).

% card_num0
tff(fact_10203_card__nat,axiom,
    finite_card_nat(top_top_set_nat) = zero_zero_nat ).

% card_nat
tff(fact_10204_card__literal,axiom,
    finite_card_literal(top_top_set_literal) = zero_zero_nat ).

% card_literal
tff(fact_10205_UNIV__bool,axiom,
    top_top_set_o = insert_o($false,insert_o($true,bot_bot_set_o)) ).

% UNIV_bool
tff(fact_10206_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_hm,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_1
tff(fact_10207_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_jc,Uu) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_nat_nat(suc,Uu)) ).

% ATP.lambda_2
tff(fact_10208_ATP_Olambda__3,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_kd,Uu)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu) = zero_zero_real ) ) ) ).

% ATP.lambda_3
tff(fact_10209_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_iw,Uu) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu)) ).

% ATP.lambda_4
tff(fact_10210_ATP_Olambda__5,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_lo,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu)),aa_real_real(sin_real,Uu)) ).

% ATP.lambda_5
tff(fact_10211_ATP_Olambda__6,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_mj,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu)),Uu) ).

% ATP.lambda_6
tff(fact_10212_ATP_Olambda__7,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_lq,Uu) = aa_real_real(aa_nat_fun_real_real(root,Uu),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_7
tff(fact_10213_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_lt,Uu) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu)) ).

% ATP.lambda_8
tff(fact_10214_ATP_Olambda__9,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_kg,Uu)
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uu) ) ).

% ATP.lambda_9
tff(fact_10215_ATP_Olambda__10,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_lh,Uu) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_ho,Uu)) ).

% ATP.lambda_10
tff(fact_10216_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_lu,Uu) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu))) ).

% ATP.lambda_11
tff(fact_10217_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_jx,Uu) = product_Pair_nat_nat(aa_nat_nat(suc,Uu)) ).

% ATP.lambda_12
tff(fact_10218_ATP_Olambda__13,axiom,
    ! [Uu: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_jd(Uu),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_13
tff(fact_10219_ATP_Olambda__14,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_14
tff(fact_10220_ATP_Olambda__15,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_15
tff(fact_10221_ATP_Olambda__16,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jf,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_16
tff(fact_10222_ATP_Olambda__17,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_kb,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(sin_real,Uua) = Uu ) ) ) ).

% ATP.lambda_17
tff(fact_10223_ATP_Olambda__18,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ka,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uua),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
        & ( aa_real_real(tan_real,Uua) = Uu ) ) ) ).

% ATP.lambda_18
tff(fact_10224_ATP_Olambda__19,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_mw,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(plus_plus_int(l2),one_one_int)) ) ).

% ATP.lambda_19
tff(fact_10225_ATP_Olambda__20,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_kf(Uu),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu) = cis(Uua) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi) ) ) ).

% ATP.lambda_20
tff(fact_10226_ATP_Olambda__21,axiom,
    ! [Uu: real,Uua: int] :
      ( aa_int_o(aTP_Lamm_km(Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu),aa_int_real(ring_1_of_int_real,aa_int_int(plus_plus_int(Uua),one_one_int))) ) ) ).

% ATP.lambda_21
tff(fact_10227_ATP_Olambda__22,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_kn(Uu),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu),aa_int_rat(ring_1_of_int_rat,aa_int_int(plus_plus_int(Uua),one_one_int))) ) ) ).

% ATP.lambda_22
tff(fact_10228_ATP_Olambda__23,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ho,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_23
tff(fact_10229_ATP_Olambda__24,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_li,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_24
tff(fact_10230_ATP_Olambda__25,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_lp(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu,Uua)) ).

% ATP.lambda_25
tff(fact_10231_ATP_Olambda__26,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jm,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_26
tff(fact_10232_ATP_Olambda__27,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ja,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_27
tff(fact_10233_ATP_Olambda__28,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_mk,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_28
tff(fact_10234_ATP_Olambda__29,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jh,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uu) ).

% ATP.lambda_29
tff(fact_10235_ATP_Olambda__30,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jg,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),Uua)),Uua) ).

% ATP.lambda_30
tff(fact_10236_ATP_Olambda__31,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_fr(Uu),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu) = one_one_complex ) ) ).

% ATP.lambda_31
tff(fact_10237_ATP_Olambda__32,axiom,
    ! [Uu: nat,Uua: real] :
      ( aa_real_o(aa_nat_fun_real_o(aTP_Lamm_fq,Uu),Uua)
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) = one_one_real ) ) ).

% ATP.lambda_32
tff(fact_10238_ATP_Olambda__33,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ma,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_33
tff(fact_10239_ATP_Olambda__34,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_mh,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_34
tff(fact_10240_ATP_Olambda__35,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ml,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu),Uua))),Uua) ).

% ATP.lambda_35
tff(fact_10241_ATP_Olambda__36,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hn,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_36
tff(fact_10242_ATP_Olambda__37,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_kc,Uu),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uua)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uua),pi)
        & ( aa_real_real(cos_real,Uua) = Uu ) ) ) ).

% ATP.lambda_37
tff(fact_10243_ATP_Olambda__38,axiom,
    ! [Uu: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_hc(Uu),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_38
tff(fact_10244_ATP_Olambda__39,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_hd(Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_39
tff(fact_10245_ATP_Olambda__40,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mg(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_lp(Uu),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_40
tff(fact_10246_ATP_Olambda__41,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_mf(Uu),Uua) = groups6591440286371151544t_real(aTP_Lamm_lp(Uu),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_41
tff(fact_10247_ATP_Olambda__42,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ke,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu)) ).

% ATP.lambda_42
tff(fact_10248_ATP_Olambda__43,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_iz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_43
tff(fact_10249_ATP_Olambda__44,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ix,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_44
tff(fact_10250_ATP_Olambda__45,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_mx,Uu),Uua) = $let(
        l2: num,
        l2:= aa_Code_integer_num(code_num_of_integer,Uu),
        $let(
          l3: num,
          l3:= aa_num_num(plus_plus_num(l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(plus_plus_num(l3),one)) ) ) ).

% ATP.lambda_45
tff(fact_10251_ATP_Olambda__46,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_my,Uu),Uua) = $let(
        l2: nat,
        l2:= aa_Code_integer_nat(code_nat_of_integer,Uu),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_46
tff(fact_10252_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jn,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uua)) ).

% ATP.lambda_47
tff(fact_10253_ATP_Olambda__48,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mb,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_48
tff(fact_10254_ATP_Olambda__49,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lz,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_49
tff(fact_10255_ATP_Olambda__50,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lw,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_50
tff(fact_10256_ATP_Olambda__51,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aTP_Lamm_fv(Uu),Uua)
    <=> aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(Uua),Uu) ) ).

% ATP.lambda_51
tff(fact_10257_ATP_Olambda__52,axiom,
    ! [Uu: set_complex,Uua: set_complex] :
      ( aa_set_complex_o(aTP_Lamm_fw(Uu),Uua)
    <=> aa_set_complex_o(ord_le211207098394363844omplex(Uua),Uu) ) ).

% ATP.lambda_52
tff(fact_10258_ATP_Olambda__53,axiom,
    ! [Uu: set_nat,Uua: set_nat] :
      ( aa_set_nat_o(aTP_Lamm_fx(Uu),Uua)
    <=> aa_set_nat_o(ord_less_eq_set_nat(Uua),Uu) ) ).

% ATP.lambda_53
tff(fact_10259_ATP_Olambda__54,axiom,
    ! [Uu: set_int,Uua: set_int] :
      ( aa_set_int_o(aTP_Lamm_fu(Uu),Uua)
    <=> aa_set_int_o(ord_less_eq_set_int(Uua),Uu) ) ).

% ATP.lambda_54
tff(fact_10260_ATP_Olambda__55,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_fs,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu) ) ).

% ATP.lambda_55
tff(fact_10261_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ki,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu) ).

% ATP.lambda_56
tff(fact_10262_ATP_Olambda__57,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ft,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_57
tff(fact_10263_ATP_Olambda__58,axiom,
    ! [Uu: assn,Uua: assn] : aa_assn_assn(aTP_Lamm_cz(Uu),Uua) = aa_assn_assn(times_times_assn(Uua),Uu) ).

% ATP.lambda_58
tff(fact_10264_ATP_Olambda__59,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_cv,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu) ).

% ATP.lambda_59
tff(fact_10265_ATP_Olambda__60,axiom,
    ! [Uu: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_cw(Uu),Uua) = aa_rat_rat(times_times_rat(Uua),Uu) ).

% ATP.lambda_60
tff(fact_10266_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cx,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu) ).

% ATP.lambda_61
tff(fact_10267_ATP_Olambda__62,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_cy(Uu),Uua) = aa_int_int(times_times_int(Uua),Uu) ).

% ATP.lambda_62
tff(fact_10268_ATP_Olambda__63,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_kl,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu) ).

% ATP.lambda_63
tff(fact_10269_ATP_Olambda__64,axiom,
    ! [Uu: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_kv,Uu),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu) ).

% ATP.lambda_64
tff(fact_10270_ATP_Olambda__65,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa_Cod1783974652533396728nteger(aTP_Lamm_kk(Uu),Uua) = plus_p5714425477246183910nteger(Uua,Uu) ).

% ATP.lambda_65
tff(fact_10271_ATP_Olambda__66,axiom,
    ! [Uu: int,Uua: int] : aa_int_int(aTP_Lamm_kj(Uu),Uua) = aa_int_int(plus_plus_int(Uua),Uu) ).

% ATP.lambda_66
tff(fact_10272_ATP_Olambda__67,axiom,
    ! [Uu: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_kx,Uu),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu) ).

% ATP.lambda_67
tff(fact_10273_ATP_Olambda__68,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_eb(Uu),Uua)
    <=> aa_complex_o(dvd_dvd_complex(Uua),Uu) ) ).

% ATP.lambda_68
tff(fact_10274_ATP_Olambda__69,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ed,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu) ) ).

% ATP.lambda_69
tff(fact_10275_ATP_Olambda__70,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ec,Uu),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu) ) ).

% ATP.lambda_70
tff(fact_10276_ATP_Olambda__71,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ji,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu) ).

% ATP.lambda_71
tff(fact_10277_ATP_Olambda__72,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lv,Uu),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu) ).

% ATP.lambda_72
tff(fact_10278_ATP_Olambda__73,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_aa(Uu),Uua)
    <=> aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uua),Uu) ) ).

% ATP.lambda_73
tff(fact_10279_ATP_Olambda__74,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ae(Uu),Uua)
    <=> member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_74
tff(fact_10280_ATP_Olambda__75,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ad(Uu),Uua)
    <=> aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_75
tff(fact_10281_ATP_Olambda__76,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_a(Uu),Uua)
    <=> aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_76
tff(fact_10282_ATP_Olambda__77,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_77
tff(fact_10283_ATP_Olambda__78,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ac(Uu),Uua)
    <=> aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_78
tff(fact_10284_ATP_Olambda__79,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_ah(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_79
tff(fact_10285_ATP_Olambda__80,axiom,
    ! [Uu: product_prod_int_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_aj(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_80
tff(fact_10286_ATP_Olambda__81,axiom,
    ! [Uu: complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_ai(Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_81
tff(fact_10287_ATP_Olambda__82,axiom,
    ! [Uu: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ak,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_82
tff(fact_10288_ATP_Olambda__83,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_al((Uu)),(Uua))
    <=> ( (Uua)
      <=> (Uu) ) ) ).

% ATP.lambda_83
tff(fact_10289_ATP_Olambda__84,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_am,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_84
tff(fact_10290_ATP_Olambda__85,axiom,
    ! [Uu: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_an,Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_85
tff(fact_10291_ATP_Olambda__86,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer] :
      ( aa_Code_integer_o(aTP_Lamm_gs(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Code_integer_nat(Uu,Uua)) ) ).

% ATP.lambda_86
tff(fact_10292_ATP_Olambda__87,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_gm(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_Pro6801748904974264397nt_nat(Uu,Uua)) ) ).

% ATP.lambda_87
tff(fact_10293_ATP_Olambda__88,axiom,
    ! [Uu: fun_complex_nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_gu(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_complex_nat(Uu,Uua)) ) ).

% ATP.lambda_88
tff(fact_10294_ATP_Olambda__89,axiom,
    ! [Uu: fun_nat_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_go(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(Uu,Uua)) ) ).

% ATP.lambda_89
tff(fact_10295_ATP_Olambda__90,axiom,
    ! [Uu: fun_int_nat,Uua: int] :
      ( aa_int_o(aTP_Lamm_gq(Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(Uu,Uua)) ) ).

% ATP.lambda_90
tff(fact_10296_ATP_Olambda__91,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_iv(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_91
tff(fact_10297_ATP_Olambda__92,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_iu(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_92
tff(fact_10298_ATP_Olambda__93,axiom,
    ! [Uu: fun_real_o,Uua: real] :
      ( aa_real_o(aTP_Lamm_mt(Uu),Uua)
    <=> aa_real_o(Uu,aa_real_real(inverse_inverse_real,Uua)) ) ).

% ATP.lambda_93
tff(fact_10299_ATP_Olambda__94,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_ku(Uu),Uua) = aa_real_real(Uu,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_94
tff(fact_10300_ATP_Olambda__95,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_in(Uu),Uua) = aa_nat_real(Uu,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_95
tff(fact_10301_ATP_Olambda__96,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_mp(Uu),Uua)
    <=> aa_nat_o(Uu,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_96
tff(fact_10302_ATP_Olambda__97,axiom,
    ! [Uu: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_ll(Uu),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_lk(Uu),Uua)) ).

% ATP.lambda_97
tff(fact_10303_ATP_Olambda__98,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_jr(Uu),Uua) = cis(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu))) ).

% ATP.lambda_98
tff(fact_10304_ATP_Olambda__99,axiom,
    ! [Uu: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_mm(Uu),Uua)
    <=> ( aa_real_real(Uu,Uua) != zero_zero_real ) ) ).

% ATP.lambda_99
tff(fact_10305_ATP_Olambda__100,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_js,Uu),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_100
tff(fact_10306_ATP_Olambda__101,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ly,Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uua)) ).

% ATP.lambda_101
tff(fact_10307_ATP_Olambda__102,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_gd(Uu),Uua)
    <=> ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uua),Uu) ) ).

% ATP.lambda_102
tff(fact_10308_ATP_Olambda__103,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_gh(Uu),Uua)
    <=> ~ member5262025264175285858nt_int(Uua,Uu) ) ).

% ATP.lambda_103
tff(fact_10309_ATP_Olambda__104,axiom,
    ! [Uu: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_gg(Uu),Uua)
    <=> ~ aa_set_complex_o(member_complex(Uua),Uu) ) ).

% ATP.lambda_104
tff(fact_10310_ATP_Olambda__105,axiom,
    ! [Uu: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_gc(Uu),Uua)
    <=> ~ aa_set_real_o(member_real(Uua),Uu) ) ).

% ATP.lambda_105
tff(fact_10311_ATP_Olambda__106,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ge(Uu),Uua)
    <=> ~ aa_set_nat_o(member_nat(Uua),Uu) ) ).

% ATP.lambda_106
tff(fact_10312_ATP_Olambda__107,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_gf(Uu),Uua)
    <=> ~ aa_set_int_o(member_int(Uua),Uu) ) ).

% ATP.lambda_107
tff(fact_10313_ATP_Olambda__108,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ls(Uu),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_108
tff(fact_10314_ATP_Olambda__109,axiom,
    ! [Uu: fun_VEBT_VEBT_nat,Uua: vEBT_VEBT] : aa_VEBT_VEBT_real(aTP_Lamm_ea(Uu),Uua) = aa_nat_real(semiri5074537144036343181t_real,aa_VEBT_VEBT_nat(Uu,Uua)) ).

% ATP.lambda_109
tff(fact_10315_ATP_Olambda__110,axiom,
    ! [Uu: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_mv(Uu),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu,Uua)) ).

% ATP.lambda_110
tff(fact_10316_ATP_Olambda__111,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_io(Uu),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_111
tff(fact_10317_ATP_Olambda__112,axiom,
    ! [Uu: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_me(Uu),Uua) = aa_real_real(cos_real,aa_nat_real(Uu,Uua)) ).

% ATP.lambda_112
tff(fact_10318_ATP_Olambda__113,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_ln(Uu),Uua) = suminf_real(aa_real_fun_nat_real(Uu,Uua)) ).

% ATP.lambda_113
tff(fact_10319_ATP_Olambda__114,axiom,
    ! [Uu: fun_Pr3238374608701176162_int_o,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_gl(Uu),Uua)
    <=> ~ aa_Pro4592068669816069595_int_o(Uu,Uua) ) ).

% ATP.lambda_114
tff(fact_10320_ATP_Olambda__115,axiom,
    ! [Uu: fun_complex_o,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_gk(Uu),Uua)
    <=> ~ aa_complex_o(Uu,Uua) ) ).

% ATP.lambda_115
tff(fact_10321_ATP_Olambda__116,axiom,
    ! [Uu: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_gi(Uu),Uua)
    <=> ~ aa_nat_o(Uu,Uua) ) ).

% ATP.lambda_116
tff(fact_10322_ATP_Olambda__117,axiom,
    ! [Uu: fun_int_o,Uua: int] :
      ( aa_int_o(aTP_Lamm_gj(Uu),Uua)
    <=> ~ aa_int_o(Uu,Uua) ) ).

% ATP.lambda_117
tff(fact_10323_ATP_Olambda__118,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_je(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_118
tff(fact_10324_ATP_Olambda__119,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_it(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_119
tff(fact_10325_ATP_Olambda__120,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_jw(Uu),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(times_3573771949741848930nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Uua),Uub)) ).

% ATP.lambda_120
tff(fact_10326_ATP_Olambda__121,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_ju(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_121
tff(fact_10327_ATP_Olambda__122,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_jv(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu)),Uub),product_Pair_int_int(aa_int_int(plus_plus_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int),aa_int_int(minus_minus_int(Uub),aa_num_int(numeral_numeral_int,Uu))),product_Pair_int_int(aa_int_int(times_times_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_122
tff(fact_10328_ATP_Olambda__123,axiom,
    ! [Uu: set_nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_ic(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_123
tff(fact_10329_ATP_Olambda__124,axiom,
    ! [Uu: set_nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_id(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_124
tff(fact_10330_ATP_Olambda__125,axiom,
    ! [Uu: set_nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ie(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_125
tff(fact_10331_ATP_Olambda__126,axiom,
    ! [Uu: set_nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_if(Uu),Uua),Uub) = $ite(aa_set_nat_o(member_nat(Uub),Uu),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_126
tff(fact_10332_ATP_Olambda__127,axiom,
    ! [Uu: nat,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hp(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_127
tff(fact_10333_ATP_Olambda__128,axiom,
    ! [Uu: nat,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hq(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_128
tff(fact_10334_ATP_Olambda__129,axiom,
    ! [Uu: nat,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_hr(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_129
tff(fact_10335_ATP_Olambda__130,axiom,
    ! [Uu: nat,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_hs(Uu),Uua),Uub) = $ite(Uub = Uu,aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_130
tff(fact_10336_ATP_Olambda__131,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_complex,Uub: nat] :
      aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hy(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_complex(Uua,Uub),zero_zero_complex) ).

% ATP.lambda_131
tff(fact_10337_ATP_Olambda__132,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hz(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_real(Uua,Uub),zero_zero_real) ).

% ATP.lambda_132
tff(fact_10338_ATP_Olambda__133,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_nat,Uub: nat] :
      aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ia(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_nat(Uua,Uub),zero_zero_nat) ).

% ATP.lambda_133
tff(fact_10339_ATP_Olambda__134,axiom,
    ! [Uu: fun_nat_o,Uua: fun_nat_int,Uub: nat] :
      aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ib(Uu),Uua),Uub) = $ite(aa_nat_o(Uu,Uub),aa_nat_int(Uua,Uub),zero_zero_int) ).

% ATP.lambda_134
tff(fact_10340_ATP_Olambda__135,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_mn(Uu),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_135
tff(fact_10341_ATP_Olambda__136,axiom,
    ! [Uu: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_lm(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu,Uub),Uua) ).

% ATP.lambda_136
tff(fact_10342_ATP_Olambda__137,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_kt(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ks(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_137
tff(fact_10343_ATP_Olambda__138,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_kr(Uu),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_kq(Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_138
tff(fact_10344_ATP_Olambda__139,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_kp(Uu),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ko(Uua),Uub),quotient_of(Uu)) ).

% ATP.lambda_139
tff(fact_10345_ATP_Olambda__140,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_la(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_140
tff(fact_10346_ATP_Olambda__141,axiom,
    ! [Uu: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_lb(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_141
tff(fact_10347_ATP_Olambda__142,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lj(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_142
tff(fact_10348_ATP_Olambda__143,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_iy(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_143
tff(fact_10349_ATP_Olambda__144,axiom,
    ! [Uu: set_VEBT_VEBTi,Uua: nat,Uub: list_VEBT_VEBTi] :
      ( aa_list_VEBT_VEBTi_o(aa_nat5816202840298588638EBTi_o(aTP_Lamm_fo(Uu),Uua),Uub)
    <=> ( ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s7982070591426661849_VEBTi(Uub)),Uua) ) ) ).

% ATP.lambda_144
tff(fact_10350_ATP_Olambda__145,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_fl(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(Uub)),Uua) ) ) ).

% ATP.lambda_145
tff(fact_10351_ATP_Olambda__146,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aa_nat2697030861539126eger_o(aTP_Lamm_fj(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3445333598471063425nteger(Uub)),Uua) ) ) ).

% ATP.lambda_146
tff(fact_10352_ATP_Olambda__147,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_fk(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s3451745648224563538omplex(Uub)),Uua) ) ) ).

% ATP.lambda_147
tff(fact_10353_ATP_Olambda__148,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( aa_list_real_o(aa_nat3629845878562903719real_o(aTP_Lamm_fm(Uu),Uua),Uub)
    <=> ( aa_set_real_o(ord_less_eq_set_real(set_real2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_real(Uub)),Uua) ) ) ).

% ATP.lambda_148
tff(fact_10354_ATP_Olambda__149,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_fn(Uu),Uua),Uub)
    <=> ( aa_set_o_o(ord_less_eq_set_o(set_o2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_o(Uub)),Uua) ) ) ).

% ATP.lambda_149
tff(fact_10355_ATP_Olambda__150,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_fp(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Uub)),Uua) ) ) ).

% ATP.lambda_150
tff(fact_10356_ATP_Olambda__151,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_fi(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Uub)),Uua) ) ) ).

% ATP.lambda_151
tff(fact_10357_ATP_Olambda__152,axiom,
    ! [Uu: set_VEBT_VEBTi,Uua: nat,Uub: list_VEBT_VEBTi] :
      ( aa_list_VEBT_VEBTi_o(aa_nat5816202840298588638EBTi_o(aTP_Lamm_fg(Uu),Uua),Uub)
    <=> ( ord_le6592769550269828683_VEBTi(set_VEBT_VEBTi2(Uub),Uu)
        & ( size_s7982070591426661849_VEBTi(Uub) = Uua ) ) ) ).

% ATP.lambda_152
tff(fact_10358_ATP_Olambda__153,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: nat,Uub: list_VEBT_VEBT] :
      ( aa_list_VEBT_VEBT_o(aa_nat1274389614848952367VEBT_o(aTP_Lamm_fd(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(ord_le4337996190870823476T_VEBT(set_VEBT_VEBT2(Uub)),Uu)
        & ( size_s6755466524823107622T_VEBT(Uub) = Uua ) ) ) ).

% ATP.lambda_153
tff(fact_10359_ATP_Olambda__154,axiom,
    ! [Uu: set_Code_integer,Uua: nat,Uub: list_Code_integer] :
      ( aa_lis2813719629158071341eger_o(aa_nat2697030861539126eger_o(aTP_Lamm_fb(Uu),Uua),Uub)
    <=> ( aa_set5643387332196575111eger_o(ord_le7084787975880047091nteger(set_Code_integer2(Uub)),Uu)
        & ( size_s3445333598471063425nteger(Uub) = Uua ) ) ) ).

% ATP.lambda_154
tff(fact_10360_ATP_Olambda__155,axiom,
    ! [Uu: set_complex,Uua: nat,Uub: list_complex] :
      ( aa_list_complex_o(aa_nat5940423528545770789plex_o(aTP_Lamm_fc(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(ord_le211207098394363844omplex(set_complex2(Uub)),Uu)
        & ( size_s3451745648224563538omplex(Uub) = Uua ) ) ) ).

% ATP.lambda_155
tff(fact_10361_ATP_Olambda__156,axiom,
    ! [Uu: set_real,Uua: nat,Uub: list_real] :
      ( aa_list_real_o(aa_nat3629845878562903719real_o(aTP_Lamm_fe(Uu),Uua),Uub)
    <=> ( aa_set_real_o(ord_less_eq_set_real(set_real2(Uub)),Uu)
        & ( size_size_list_real(Uub) = Uua ) ) ) ).

% ATP.lambda_156
tff(fact_10362_ATP_Olambda__157,axiom,
    ! [Uu: set_o,Uua: nat,Uub: list_o] :
      ( aa_list_o_o(aa_nat_fun_list_o_o(aTP_Lamm_ff(Uu),Uua),Uub)
    <=> ( aa_set_o_o(ord_less_eq_set_o(set_o2(Uub)),Uu)
        & ( size_size_list_o(Uub) = Uua ) ) ) ).

% ATP.lambda_157
tff(fact_10363_ATP_Olambda__158,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_fh(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(ord_less_eq_set_nat(set_nat2(Uub)),Uu)
        & ( size_size_list_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_158
tff(fact_10364_ATP_Olambda__159,axiom,
    ! [Uu: set_int,Uua: nat,Uub: list_int] :
      ( aa_list_int_o(aa_nat3638697132715757351_int_o(aTP_Lamm_fa(Uu),Uua),Uub)
    <=> ( aa_set_int_o(ord_less_eq_set_int(set_int2(Uub)),Uu)
        & ( size_size_list_int(Uub) = Uua ) ) ) ).

% ATP.lambda_159
tff(fact_10365_ATP_Olambda__160,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_na(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(aa_nat_nat(suc,Uub)),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_160
tff(fact_10366_ATP_Olambda__161,axiom,
    ! [Uu: list_VEBT_VEBT,Uua: array_VEBT_VEBTi,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_arr7909941341466502432i_assn(aTP_Lamm_eq(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uua),Uub)),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uu),Uub)) ).

% ATP.lambda_161
tff(fact_10367_ATP_Olambda__162,axiom,
    ! [Uu: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_mz(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_162
tff(fact_10368_ATP_Olambda__163,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jj(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uub)) ).

% ATP.lambda_163
tff(fact_10369_ATP_Olambda__164,axiom,
    ! [Uu: vEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_bt(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_164
tff(fact_10370_ATP_Olambda__165,axiom,
    ! [Uu: product_prod_int_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_bx(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_165
tff(fact_10371_ATP_Olambda__166,axiom,
    ! [Uu: complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_bw(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_166
tff(fact_10372_ATP_Olambda__167,axiom,
    ! [Uu: real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_bs(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_167
tff(fact_10373_ATP_Olambda__168,axiom,
    ! [Uu: $o,Uua: set_o,Uub: $o] :
      ( aa_o_o(aa_set_o_fun_o_o(aTP_Lamm_br((Uu)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu) )
        | aa_set_o_o(member_o((Uub)),Uua) ) ) ).

% ATP.lambda_168
tff(fact_10374_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_bu(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_169
tff(fact_10375_ATP_Olambda__170,axiom,
    ! [Uu: int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_bv(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_170
tff(fact_10376_ATP_Olambda__171,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_jz(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_171
tff(fact_10377_ATP_Olambda__172,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_jy(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_172
tff(fact_10378_ATP_Olambda__173,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ga(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_173
tff(fact_10379_ATP_Olambda__174,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_fz(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_174
tff(fact_10380_ATP_Olambda__175,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_fy(Uu),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_175
tff(fact_10381_ATP_Olambda__176,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_af(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uub) ) ) ).

% ATP.lambda_176
tff(fact_10382_ATP_Olambda__177,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ag(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu),Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_177
tff(fact_10383_ATP_Olambda__178,axiom,
    ! [Uu: set_Code_integer,Uua: set_Code_integer,Uub: set_Code_integer] :
      ( aa_set5643387332196575111eger_o(aa_set4044230130517860621eger_o(aTP_Lamm_ev(Uu),Uua),Uub)
    <=> ( minus_2355218937544613996nteger(Uub,Uu) = minus_2355218937544613996nteger(Uua,Uu) ) ) ).

% ATP.lambda_178
tff(fact_10384_ATP_Olambda__179,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: set_complex] :
      ( aa_set_complex_o(aa_set8643480048548153067plex_o(aTP_Lamm_ew(Uu),Uua),Uub)
    <=> ( minus_811609699411566653omplex(Uub,Uu) = minus_811609699411566653omplex(Uua,Uu) ) ) ).

% ATP.lambda_179
tff(fact_10385_ATP_Olambda__180,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(aTP_Lamm_ex(Uu),Uua),Uub)
    <=> ( minus_minus_set_nat(Uub,Uu) = minus_minus_set_nat(Uua,Uu) ) ) ).

% ATP.lambda_180
tff(fact_10386_ATP_Olambda__181,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: set_int] :
      ( aa_set_int_o(aa_set5425576183772390127_int_o(aTP_Lamm_eu(Uu),Uua),Uub)
    <=> ( minus_minus_set_int(Uub,Uu) = minus_minus_set_int(Uua,Uu) ) ) ).

% ATP.lambda_181
tff(fact_10387_ATP_Olambda__182,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_bg(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_182
tff(fact_10388_ATP_Olambda__183,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_bj(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_183
tff(fact_10389_ATP_Olambda__184,axiom,
    ! [Uu: set_complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_bi(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_184
tff(fact_10390_ATP_Olambda__185,axiom,
    ! [Uu: set_real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bf(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_185
tff(fact_10391_ATP_Olambda__186,axiom,
    ! [Uu: set_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bk(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_186
tff(fact_10392_ATP_Olambda__187,axiom,
    ! [Uu: set_int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_bh(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_187
tff(fact_10393_ATP_Olambda__188,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_ay(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_188
tff(fact_10394_ATP_Olambda__189,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_ba(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_189
tff(fact_10395_ATP_Olambda__190,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_az(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_190
tff(fact_10396_ATP_Olambda__191,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bb(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_191
tff(fact_10397_ATP_Olambda__192,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_bc((Uu)),Uua),(Uub))
    <=> ( ( (Uu)
        <=> (Uub) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_192
tff(fact_10398_ATP_Olambda__193,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_bd(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_193
tff(fact_10399_ATP_Olambda__194,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_be(Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_194
tff(fact_10400_ATP_Olambda__195,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_ar(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_195
tff(fact_10401_ATP_Olambda__196,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_at(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_196
tff(fact_10402_ATP_Olambda__197,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_as(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_197
tff(fact_10403_ATP_Olambda__198,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_au(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_198
tff(fact_10404_ATP_Olambda__199,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_av((Uu)),Uua),(Uub))
    <=> ( ( (Uub)
        <=> (Uu) )
        & aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_199
tff(fact_10405_ATP_Olambda__200,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_aw(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_200
tff(fact_10406_ATP_Olambda__201,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_ax(Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_201
tff(fact_10407_ATP_Olambda__202,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: $o] : aa_o_assn(aa_VEB6687300313434771872o_assn(aTP_Lamm_do(Uu),Uua),(Uub)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,(Uub) = aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Uu))) ).

% ATP.lambda_202
tff(fact_10408_ATP_Olambda__203,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_dr(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,Uub = vEBT_vebt_mint(Uu))) ).

% ATP.lambda_203
tff(fact_10409_ATP_Olambda__204,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option_nat] : aa_option_nat_assn(aa_VEB7611521244314475576t_assn(aTP_Lamm_ds(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,Uub = vEBT_vebt_maxt(Uu))) ).

% ATP.lambda_204
tff(fact_10410_ATP_Olambda__205,axiom,
    ! [Uu: set_VEBT_VEBT,Uua: set_VEBT_VEBT,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_set2497645856053385599VEBT_o(aTP_Lamm_bm(Uu),Uua),Uub)
    <=> ( aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uu)
        & ~ aa_set_VEBT_VEBT_o(member_VEBT_VEBT(Uub),Uua) ) ) ).

% ATP.lambda_205
tff(fact_10411_ATP_Olambda__206,axiom,
    ! [Uu: set_Pr958786334691620121nt_int,Uua: set_Pr958786334691620121nt_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_set2224860389584585407_int_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> ( member5262025264175285858nt_int(Uub,Uu)
        & ~ member5262025264175285858nt_int(Uub,Uua) ) ) ).

% ATP.lambda_206
tff(fact_10412_ATP_Olambda__207,axiom,
    ! [Uu: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_bo(Uu),Uua),Uub)
    <=> ( aa_set_complex_o(member_complex(Uub),Uu)
        & ~ aa_set_complex_o(member_complex(Uub),Uua) ) ) ).

% ATP.lambda_207
tff(fact_10413_ATP_Olambda__208,axiom,
    ! [Uu: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_bl(Uu),Uua),Uub)
    <=> ( aa_set_real_o(member_real(Uub),Uu)
        & ~ aa_set_real_o(member_real(Uub),Uua) ) ) ).

% ATP.lambda_208
tff(fact_10414_ATP_Olambda__209,axiom,
    ! [Uu: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_bq(Uu),Uua),Uub)
    <=> ( aa_set_nat_o(member_nat(Uub),Uu)
        & ~ aa_set_nat_o(member_nat(Uub),Uua) ) ) ).

% ATP.lambda_209
tff(fact_10415_ATP_Olambda__210,axiom,
    ! [Uu: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_bn(Uu),Uua),Uub)
    <=> ( aa_set_int_o(member_int(Uub),Uu)
        & ~ aa_set_int_o(member_int(Uub),Uua) ) ) ).

% ATP.lambda_210
tff(fact_10416_ATP_Olambda__211,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_kh(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu) ) ).

% ATP.lambda_211
tff(fact_10417_ATP_Olambda__212,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_gb(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu) = Uua ) ) ).

% ATP.lambda_212
tff(fact_10418_ATP_Olambda__213,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_jt(Uu),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu ) ) ).

% ATP.lambda_213
tff(fact_10419_ATP_Olambda__214,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_md(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_214
tff(fact_10420_ATP_Olambda__215,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lk(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_215
tff(fact_10421_ATP_Olambda__216,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ip(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_216
tff(fact_10422_ATP_Olambda__217,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_jo(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_217
tff(fact_10423_ATP_Olambda__218,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ey(Uu),Uua),Uub)
    <=> ( aa_nat_o(Uu,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_218
tff(fact_10424_ATP_Olambda__219,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mu(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_219
tff(fact_10425_ATP_Olambda__220,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mo(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu,Uub)) ).

% ATP.lambda_220
tff(fact_10426_ATP_Olambda__221,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ir(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_221
tff(fact_10427_ATP_Olambda__222,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_lr(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_222
tff(fact_10428_ATP_Olambda__223,axiom,
    ! [Uu: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ii(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(Uu,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_223
tff(fact_10429_ATP_Olambda__224,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_ij(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_224
tff(fact_10430_ATP_Olambda__225,axiom,
    ! [Uu: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_ik(Uu),Uua),Uub) = aa_int_int(plus_plus_int(aa_nat_int(Uu,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_225
tff(fact_10431_ATP_Olambda__226,axiom,
    ! [Uu: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_kz(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_226
tff(fact_10432_ATP_Olambda__227,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ez(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu,Uub)),Uua) ) ).

% ATP.lambda_227
tff(fact_10433_ATP_Olambda__228,axiom,
    ! [Uu: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_hw(Uu),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu,Uub)),Uua) ).

% ATP.lambda_228
tff(fact_10434_ATP_Olambda__229,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_hx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_229
tff(fact_10435_ATP_Olambda__230,axiom,
    ! [Uu: fun_VEBT_VEBTi_assn,Uua: assn,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_ass4389913640333755968i_assn(aTP_Lamm_db(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(Uu,Uub)),Uua) ).

% ATP.lambda_230
tff(fact_10436_ATP_Olambda__231,axiom,
    ! [Uu: fun_option_nat_assn,Uua: assn,Uub: option_nat] : aa_option_nat_assn(aa_ass4112278883933122603t_assn(aTP_Lamm_dc(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_option_nat_assn(Uu,Uub)),Uua) ).

% ATP.lambda_231
tff(fact_10437_ATP_Olambda__232,axiom,
    ! [Uu: fun_li3352006551604695530i_assn,Uua: assn,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_ass1897699608592665680i_assn(aTP_Lamm_et(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(Uu,Uub)),Uua) ).

% ATP.lambda_232
tff(fact_10438_ATP_Olambda__233,axiom,
    ! [Uu: fun_o_assn,Uua: assn,Uub: $o] : aa_o_assn(aa_assn_fun_o_assn(aTP_Lamm_da(Uu),Uua),(Uub)) = aa_assn_assn(times_times_assn(aa_o_assn(Uu,(Uub))),Uua) ).

% ATP.lambda_233
tff(fact_10439_ATP_Olambda__234,axiom,
    ! [Uu: fun_nat_assn,Uua: assn,Uub: nat] : aa_nat_assn(aa_assn_fun_nat_assn(aTP_Lamm_dd(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(aa_nat_assn(Uu,Uub)),Uua) ).

% ATP.lambda_234
tff(fact_10440_ATP_Olambda__235,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_il(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu,Uub)),Uua) ).

% ATP.lambda_235
tff(fact_10441_ATP_Olambda__236,axiom,
    ! [Uu: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_kw(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_236
tff(fact_10442_ATP_Olambda__237,axiom,
    ! [Uu: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_mi(Uu),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu) ).

% ATP.lambda_237
tff(fact_10443_ATP_Olambda__238,axiom,
    ! [Uu: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ky(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu,Uub)),Uua) ).

% ATP.lambda_238
tff(fact_10444_ATP_Olambda__239,axiom,
    ! [Uu: vEBT_VEBT,Uua: fun_VEBT_VEBT_o,Uub: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aa_fun8193769468198689164VEBT_o(aTP_Lamm_by(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_VEBT_VEBT_o(Uua,Uub) ) ) ).

% ATP.lambda_239
tff(fact_10445_ATP_Olambda__240,axiom,
    ! [Uu: product_prod_int_int,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_ce(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_Pro4592068669816069595_int_o(Uua,Uub) ) ) ).

% ATP.lambda_240
tff(fact_10446_ATP_Olambda__241,axiom,
    ! [Uu: complex,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_cd(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_complex_o(Uua,Uub) ) ) ).

% ATP.lambda_241
tff(fact_10447_ATP_Olambda__242,axiom,
    ! [Uu: real,Uua: fun_real_o,Uub: real] :
      ( aa_real_o(aa_fun5868358341814070780real_o(aTP_Lamm_bz(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_real_o(Uua,Uub) ) ) ).

% ATP.lambda_242
tff(fact_10448_ATP_Olambda__243,axiom,
    ! [Uu: $o,Uua: fun_o_o,Uub: $o] :
      ( aa_o_o(aa_fun_o_o_fun_o_o(aTP_Lamm_ca((Uu)),Uua),(Uub))
    <=> ( ~ ( (Uub)
          <=> (Uu) )
       => aa_o_o(Uua,(Uub)) ) ) ).

% ATP.lambda_243
tff(fact_10449_ATP_Olambda__244,axiom,
    ! [Uu: nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_cb(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_244
tff(fact_10450_ATP_Olambda__245,axiom,
    ! [Uu: int,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_cc(Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa_int_o(Uua,Uub) ) ) ).

% ATP.lambda_245
tff(fact_10451_ATP_Olambda__246,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_lx(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu),Uub)) ).

% ATP.lambda_246
tff(fact_10452_ATP_Olambda__247,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_iq(Uu),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_247
tff(fact_10453_ATP_Olambda__248,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ms(Uu),Uua),Uub)
    <=> aa_set_real_o(member_real(Uub),set_or1633881224788618240n_real(Uu,Uua)) ) ).

% ATP.lambda_248
tff(fact_10454_ATP_Olambda__249,axiom,
    ! [Uu: complex,Uua: fun_nat_complex,Uub: nat] : aa_nat_complex(aa_fun2955221450537884980omplex(aTP_Lamm_hu(Uu),Uua),Uub) = aa_complex_complex(times_times_complex(Uu),aa_nat_complex(Uua,Uub)) ).

% ATP.lambda_249
tff(fact_10455_ATP_Olambda__250,axiom,
    ! [Uu: assn,Uua: fun_li3352006551604695530i_assn,Uub: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aa_fun6795544665460849810i_assn(aTP_Lamm_es(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uu),aa_lis2508312872377092907i_assn(Uua,Uub)) ).

% ATP.lambda_250
tff(fact_10456_ATP_Olambda__251,axiom,
    ! [Uu: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_hv(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_251
tff(fact_10457_ATP_Olambda__252,axiom,
    ! [Uu: fun_VEBT_VEBTi_assn,Uua: assn,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_ass4389913640333755968i_assn(aTP_Lamm_en(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uua),aa_VEBT_VEBTi_assn(Uu,Uub)) ).

% ATP.lambda_252
tff(fact_10458_ATP_Olambda__253,axiom,
    ! [Uu: fun_option_nat_assn,Uua: assn,Uub: option_nat] : aa_option_nat_assn(aa_ass4112278883933122603t_assn(aTP_Lamm_eo(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uua),aa_option_nat_assn(Uu,Uub)) ).

% ATP.lambda_253
tff(fact_10459_ATP_Olambda__254,axiom,
    ! [Uu: fun_o_assn,Uua: assn,Uub: $o] : aa_o_assn(aa_assn_fun_o_assn(aTP_Lamm_em(Uu),Uua),(Uub)) = aa_assn_assn(times_times_assn(Uua),aa_o_assn(Uu,(Uub))) ).

% ATP.lambda_254
tff(fact_10460_ATP_Olambda__255,axiom,
    ! [Uu: fun_nat_assn,Uua: assn,Uub: nat] : aa_nat_assn(aa_assn_fun_nat_assn(aTP_Lamm_ep(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uua),aa_nat_assn(Uu,Uub)) ).

% ATP.lambda_255
tff(fact_10461_ATP_Olambda__256,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_im(Uu),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),aa_nat_real(Uu,Uub)) ).

% ATP.lambda_256
tff(fact_10462_ATP_Olambda__257,axiom,
    ! [Uu: assn,Uua: vEBT_VEBTi,Uub: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_VEB1545114916007340237i_assn(aTP_Lamm_ej(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_257
tff(fact_10463_ATP_Olambda__258,axiom,
    ! [Uu: assn,Uua: option_nat,Uub: option_nat] : aa_option_nat_assn(aa_opt3861602675778688675t_assn(aTP_Lamm_ek(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_258
tff(fact_10464_ATP_Olambda__259,axiom,
    ! [Uu: assn,Uua: $o,Uub: $o] : aa_o_assn(aa_o_fun_o_assn(aTP_Lamm_ei(Uu),(Uua)),(Uub)) = aa_assn_assn(times_times_assn(Uu),aa_o_assn(pure_assn,(Uub) = (Uua))) ).

% ATP.lambda_259
tff(fact_10465_ATP_Olambda__260,axiom,
    ! [Uu: assn,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_el(Uu),Uua),Uub) = aa_assn_assn(times_times_assn(Uu),aa_o_assn(pure_assn,Uub = Uua)) ).

% ATP.lambda_260
tff(fact_10466_ATP_Olambda__261,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: code_integer,Uub: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_gz(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uu,Uub))) ) ).

% ATP.lambda_261
tff(fact_10467_ATP_Olambda__262,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: product_prod_int_int,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(aTP_Lamm_gw(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_Pro6801748904974264397nt_nat(Uu,Uub)),aa_Pro6801748904974264397nt_nat(Uu,Uub))) ) ).

% ATP.lambda_262
tff(fact_10468_ATP_Olambda__263,axiom,
    ! [Uu: fun_complex_nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_ha(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_complex_nat(Uu,Uub)),aa_complex_nat(Uu,Uub))) ) ).

% ATP.lambda_263
tff(fact_10469_ATP_Olambda__264,axiom,
    ! [Uu: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_gx(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uu,Uub))) ) ).

% ATP.lambda_264
tff(fact_10470_ATP_Olambda__265,axiom,
    ! [Uu: fun_int_nat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_gy(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(Uub = Uua,aa_nat_nat(suc,aa_int_nat(Uu,Uub)),aa_int_nat(Uu,Uub))) ) ).

% ATP.lambda_265
tff(fact_10471_ATP_Olambda__266,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_o,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun8214818745163655228eger_o(aTP_Lamm_hh(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_Code_integer_o(Uua,Uub),aa_Code_integer_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_266
tff(fact_10472_ATP_Olambda__267,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: fun_Pr3238374608701176162_int_o,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aTP_Lamm_he(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_Pro4592068669816069595_int_o(Uua,Uub),aa_Pro6801748904974264397nt_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_267
tff(fact_10473_ATP_Olambda__268,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_o,Uub: complex] :
      ( aa_complex_o(aa_fun2514317545224092796plex_o(aTP_Lamm_hi(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_complex_o(Uua,Uub),aa_complex_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_268
tff(fact_10474_ATP_Olambda__269,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_o,Uub: nat] :
      ( aa_nat_o(aa_fun5937193402907943164_nat_o(aTP_Lamm_hf(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_nat_o(Uua,Uub),aa_nat_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_269
tff(fact_10475_ATP_Olambda__270,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_o,Uub: int] :
      ( aa_int_o(aa_fun6243517787477317628_int_o(aTP_Lamm_hg(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),
          $ite(aa_int_o(Uua,Uub),aa_int_nat(Uu,Uub),zero_zero_nat)) ) ).

% ATP.lambda_270
tff(fact_10476_ATP_Olambda__271,axiom,
    ! [Uu: fun_Code_integer_nat,Uua: fun_Code_integer_nat,Uub: code_integer] :
      ( aa_Code_integer_o(aa_fun4054834855839773292eger_o(aTP_Lamm_gt(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Code_integer_nat(Uu,Uub)),aa_Code_integer_nat(Uua,Uub))) ) ).

% ATP.lambda_271
tff(fact_10477_ATP_Olambda__272,axiom,
    ! [Uu: fun_Pr5160692531401422620nt_nat,Uua: fun_Pr5160692531401422620nt_nat,Uub: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun6302685600795359042_int_o(aTP_Lamm_gn(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_Pro6801748904974264397nt_nat(Uu,Uub)),aa_Pro6801748904974264397nt_nat(Uua,Uub))) ) ).

% ATP.lambda_272
tff(fact_10478_ATP_Olambda__273,axiom,
    ! [Uu: fun_complex_nat,Uua: fun_complex_nat,Uub: complex] :
      ( aa_complex_o(aa_fun8723867720292580170plex_o(aTP_Lamm_gv(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_complex_nat(Uu,Uub)),aa_complex_nat(Uua,Uub))) ) ).

% ATP.lambda_273
tff(fact_10479_ATP_Olambda__274,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] :
      ( aa_nat_o(aa_fun8673969331427483654_nat_o(aTP_Lamm_gp(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(Uu,Uub)),aa_nat_nat(Uua,Uub))) ) ).

% ATP.lambda_274
tff(fact_10480_ATP_Olambda__275,axiom,
    ! [Uu: fun_int_nat,Uua: fun_int_nat,Uub: int] :
      ( aa_int_o(aa_fun5425778157628386638_int_o(aTP_Lamm_gr(Uu),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(Uu,Uub)),aa_int_nat(Uua,Uub))) ) ).

% ATP.lambda_275
tff(fact_10481_ATP_Olambda__276,axiom,
    ! [Uu: fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_mr(Uu),Uua),Uub)
    <=> aa_real_o(Uu,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua)) ) ).

% ATP.lambda_276
tff(fact_10482_ATP_Olambda__277,axiom,
    ! [Uu: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ht(Uu),Uua),Uub) = aa_nat_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_277
tff(fact_10483_ATP_Olambda__278,axiom,
    ! [Uu: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_mq(Uu),Uua),Uub)
    <=> aa_nat_o(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_278
tff(fact_10484_ATP_Olambda__279,axiom,
    ! [Uu: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_mc(Uu),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu,Uub)),Uua)) ).

% ATP.lambda_279
tff(fact_10485_ATP_Olambda__280,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_dz(Uu),Uua),Uub) = aa_o_assn(pure_assn,Uub = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Uu),Uua)) ).

% ATP.lambda_280
tff(fact_10486_ATP_Olambda__281,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa_nat_assn(aa_nat_fun_nat_assn(aTP_Lamm_dy(Uu),Uua),Uub) = aa_o_assn(pure_assn,Uub = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Uu),Uua)) ).

% ATP.lambda_281
tff(fact_10487_ATP_Olambda__282,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_jq(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_jp(Uu),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_282
tff(fact_10488_ATP_Olambda__283,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_lg(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_283
tff(fact_10489_ATP_Olambda__284,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_le(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_284
tff(fact_10490_ATP_Olambda__285,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_lc(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_285
tff(fact_10491_ATP_Olambda__286,axiom,
    ! [Uu: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ld(Uu),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_286
tff(fact_10492_ATP_Olambda__287,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jl(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_287
tff(fact_10493_ATP_Olambda__288,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_jk(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_288
tff(fact_10494_ATP_Olambda__289,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_kq(Uu),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(times_times_int(Uu),Uuc)),aa_int_int(times_times_int(Uua),Uub)) ) ).

% ATP.lambda_289
tff(fact_10495_ATP_Olambda__290,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: $o] : aa_o_assn(aa_nat_fun_o_assn(aa_VEB8176250900481067561o_assn(aTP_Lamm_ao(Uu),Uua),Uub),(Uuc)) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,(Uuc) = aa_nat_o(aa_VEB2314620471664397097_nat_o(vEBT_vebt_member,Uu),Uub))) ).

% ATP.lambda_290
tff(fact_10496_ATP_Olambda__291,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option_nat] : aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_ap(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,Uuc = vEBT_vebt_succ(Uu,Uub))) ).

% ATP.lambda_291
tff(fact_10497_ATP_Olambda__292,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option_nat] : aa_option_nat_assn(aa_nat435388790147128531t_assn(aa_VEB1697702717351837573t_assn(aTP_Lamm_aq(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,Uu),Uua)),aa_o_assn(pure_assn,Uuc = vEBT_vebt_pred(Uu,Uub))) ).

% ATP.lambda_292
tff(fact_10498_ATP_Olambda__293,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBTi,Uub: array_VEBT_VEBTi,Uuc: vEBT_VEBTi] : aa_VEBT_VEBTi_assn(aa_arr8081119733880211216i_assn(aa_lis5880058344902901532i_assn(aTP_Lamm_ef(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(Uub),Uua)),aa_o_assn(pure_assn,Uuc = nth_VEBT_VEBTi(Uua,Uu))) ).

% ATP.lambda_293
tff(fact_10499_ATP_Olambda__294,axiom,
    ! [Uu: nat,Uua: list_option_nat,Uub: array_option_nat,Uuc: option_nat] : aa_option_nat_assn(aa_arr7483714166967448038t_assn(aa_lis7582702936096254407t_assn(aTP_Lamm_eg(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(snga_assn_option_nat(Uub,Uua)),aa_o_assn(pure_assn,Uuc = nth_option_nat(Uua,Uu))) ).

% ATP.lambda_294
tff(fact_10500_ATP_Olambda__295,axiom,
    ! [Uu: nat,Uua: list_o,Uub: array_o,Uuc: $o] : aa_o_assn(aa_arr2544151649598861476o_assn(aa_lis966790859614740569o_assn(aTP_Lamm_ee(Uu),Uua),Uub),(Uuc)) = aa_assn_assn(times_times_assn(snga_assn_o(Uub,Uua)),aa_o_assn(pure_assn,(Uuc) = aa_nat_o(nth_o(Uua),Uu))) ).

% ATP.lambda_295
tff(fact_10501_ATP_Olambda__296,axiom,
    ! [Uu: nat,Uua: list_nat,Uub: array_nat,Uuc: nat] : aa_nat_assn(aa_arr3838850077245469638t_assn(aa_lis668879630561721847t_assn(aTP_Lamm_eh(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(snga_assn_nat(Uub,Uua)),aa_o_assn(pure_assn,Uuc = aa_nat_nat(nth_nat(Uua),Uu))) ).

% ATP.lambda_296
tff(fact_10502_ATP_Olambda__297,axiom,
    ! [Uu: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_jp(Uu),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_297
tff(fact_10503_ATP_Olambda__298,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_dp(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_o_assn(pure_assn,Uuc = vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uu)))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_298
tff(fact_10504_ATP_Olambda__299,axiom,
    ! [Uu: nat,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBTi,Uuc: option_nat] : aa_option_nat_assn(aa_lis7387135221810377032t_assn(aa_lis7458637400188059141t_assn(aTP_Lamm_dq(Uu),Uua),Uub),Uuc) = aa_assn_assn(times_times_assn(aa_o_assn(pure_assn,Uuc = vEBT_vebt_maxt(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(Uua),Uu)))),aa_lis2508312872377092907i_assn(vEBT_L6296928887356842470_VEBTi(vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_299
tff(fact_10505_ATP_Olambda__300,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ko(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(minus_minus_int(aa_int_int(times_times_int(Uu),Uuc)),aa_int_int(times_times_int(Uub),Uua)),aa_int_int(times_times_int(Uua),Uuc))) ).

% ATP.lambda_300
tff(fact_10506_ATP_Olambda__301,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_ks(Uu),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(times_times_int(Uu),Uuc),aa_int_int(times_times_int(Uua),Uub))) ).

% ATP.lambda_301
tff(fact_10507_ATP_Olambda__302,axiom,
    ! [Uu: fun_VE1571388752190420990T_assn,Uua: list_VEBT_VEBT,Uub: list_VEBT_VEBT,Uuc: fun_o_assn,Uud: $o] : aa_o_assn(aa_fun4536517337817438332o_assn(aa_lis1936063806678856193o_assn(aa_lis5927258619744804946o_assn(aTP_Lamm_de(Uu),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(times_times_assn(vEBT_L1279224858307276611T_VEBT(Uu,Uua,Uub)),aa_o_assn(Uuc,(Uud))) ).

% ATP.lambda_302
tff(fact_10508_ATP_Olambda__303,axiom,
    ! [Uu: fun_VE4931415397359990460t_assn,Uua: list_VEBT_VEBT,Uub: list_nat,Uuc: fun_o_assn,Uud: $o] : aa_o_assn(aa_fun4536517337817438332o_assn(aa_lis3456267157703431713o_assn(aa_lis1555086301578635742o_assn(aTP_Lamm_df(Uu),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(times_times_assn(vEBT_L8296926524756676353BT_nat(Uu,Uua,Uub)),aa_o_assn(Uuc,(Uud))) ).

% ATP.lambda_303
tff(fact_10509_ATP_Olambda__304,axiom,
    ! [Uu: fun_re560840401387337987i_assn,Uua: list_real,Uub: list_VEBT_VEBTi,Uuc: fun_o_assn,Uud: $o] : aa_o_assn(aa_fun4536517337817438332o_assn(aa_lis522499054687323654o_assn(aa_lis3287929508098246591o_assn(aTP_Lamm_dh(Uu),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L9060850011106065574_VEBTi(Uu,Uua),Uub)),aa_o_assn(Uuc,(Uud))) ).

% ATP.lambda_304
tff(fact_10510_ATP_Olambda__305,axiom,
    ! [Uu: fun_re8050108611700907676T_assn,Uua: list_real,Uub: list_VEBT_VEBT,Uuc: fun_o_assn,Uud: $o] : aa_o_assn(aa_fun4536517337817438332o_assn(aa_lis1936063806678856193o_assn(aa_lis9036567463675750230o_assn(aTP_Lamm_dg(Uu),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(times_times_assn(vEBT_L4595930785310033027T_VEBT(Uu,Uua,Uub)),aa_o_assn(Uuc,(Uud))) ).

% ATP.lambda_305
tff(fact_10511_ATP_Olambda__306,axiom,
    ! [Uu: fun_re574865750389657758t_assn,Uua: list_real,Uub: list_nat,Uuc: fun_o_assn,Uud: $o] : aa_o_assn(aa_fun4536517337817438332o_assn(aa_lis3456267157703431713o_assn(aa_lis7556923739995607514o_assn(aTP_Lamm_di(Uu),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(times_times_assn(vEBT_L1446010312343316929al_nat(Uu,Uua,Uub)),aa_o_assn(Uuc,(Uud))) ).

% ATP.lambda_306
tff(fact_10512_ATP_Olambda__307,axiom,
    ! [Uu: fun_o_9029690514724572677i_assn,Uua: list_o,Uub: list_VEBT_VEBTi,Uuc: fun_o_assn,Uud: $o] : aa_o_assn(aa_fun4536517337817438332o_assn(aa_lis522499054687323654o_assn(aa_lis958962040488458791o_assn(aTP_Lamm_dk(Uu),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L3704169666673096010_VEBTi(Uu,Uua),Uub)),aa_o_assn(Uuc,(Uud))) ).

% ATP.lambda_307
tff(fact_10513_ATP_Olambda__308,axiom,
    ! [Uu: fun_o_4353168661238401050T_assn,Uua: list_o,Uub: list_VEBT_VEBT,Uuc: fun_o_assn,Uud: $o] : aa_o_assn(aa_fun4536517337817438332o_assn(aa_lis1936063806678856193o_assn(aa_lis6424289830231647982o_assn(aTP_Lamm_dj(Uu),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(times_times_assn(vEBT_L1750719106661372127T_VEBT(Uu,Uua,Uub)),aa_o_assn(Uuc,(Uud))) ).

% ATP.lambda_308
tff(fact_10514_ATP_Olambda__309,axiom,
    ! [Uu: fun_o_fun_nat_assn,Uua: list_o,Uub: list_nat,Uuc: fun_o_assn,Uud: $o] : aa_o_assn(aa_fun4536517337817438332o_assn(aa_lis3456267157703431713o_assn(aa_lis2327745721547492418o_assn(aTP_Lamm_dl(Uu),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(times_times_assn(vEBT_L4785011123346445925_o_nat(Uu,Uua,Uub)),aa_o_assn(Uuc,(Uud))) ).

% ATP.lambda_309
tff(fact_10515_ATP_Olambda__310,axiom,
    ! [Uu: fun_na6467670216208870567i_assn,Uua: list_nat,Uub: list_VEBT_VEBTi,Uuc: fun_o_assn,Uud: $o] : aa_o_assn(aa_fun4536517337817438332o_assn(aa_lis522499054687323654o_assn(aa_lis4461610605147045859o_assn(aTP_Lamm_dn(Uu),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(times_times_assn(aa_lis2508312872377092907i_assn(vEBT_L4387162340545308618_VEBTi(Uu,Uua),Uub)),aa_o_assn(Uuc,(Uud))) ).

% ATP.lambda_310
tff(fact_10516_ATP_Olambda__311,axiom,
    ! [Uu: fun_na4489259242115493880T_assn,Uua: list_nat,Uub: list_VEBT_VEBT,Uuc: fun_o_assn,Uud: $o] : aa_o_assn(aa_fun4536517337817438332o_assn(aa_lis1936063806678856193o_assn(aa_lis765023829625672754o_assn(aTP_Lamm_dm(Uu),Uua),Uub),Uuc),(Uud)) = aa_assn_assn(times_times_assn(vEBT_L8158188754432654943T_VEBT(Uu,Uua,Uub)),aa_o_assn(Uuc,(Uud))) ).

% ATP.lambda_311
tff(fact_10517_ATP_Olambda__312,axiom,
    ! [Uu: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_lf(Uu),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_le(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),Uuc)))))) ).

% ATP.lambda_312
tff(fact_10518_ATP_Olambda__313,axiom,
    ! [Uu: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_ig(Uu),Uua) = Uu ).

% ATP.lambda_313
tff(fact_10519_ATP_Olambda__314,axiom,
    ! [Uu: assn,Uua: list_VEBT_VEBTi] : aa_lis2508312872377092907i_assn(aTP_Lamm_er(Uu),Uua) = Uu ).

% ATP.lambda_314
tff(fact_10520_ATP_Olambda__315,axiom,
    ! [Uu: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ih,Uu),Uua) = Uu ).

% ATP.lambda_315
tff(fact_10521_ATP_Olambda__316,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_is,Uu) = Uu ).

% ATP.lambda_316
tff(fact_10522_ATP_Olambda__317,axiom,
    ! [Uu: assn] : aa_assn_assn(aTP_Lamm_cp,Uu) = Uu ).

% ATP.lambda_317
tff(fact_10523_ATP_Olambda__318,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_cl,Uu) = Uu ).

% ATP.lambda_318
tff(fact_10524_ATP_Olambda__319,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_cm,Uu) = Uu ).

% ATP.lambda_319
tff(fact_10525_ATP_Olambda__320,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_cn,Uu) = Uu ).

% ATP.lambda_320
tff(fact_10526_ATP_Olambda__321,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_co,Uu) = Uu ).

% ATP.lambda_321
tff(fact_10527_ATP_Olambda__322,axiom,
    ! [Uu: complex] : aa_complex_complex(aTP_Lamm_cq,Uu) = zero_zero_complex ).

% ATP.lambda_322
tff(fact_10528_ATP_Olambda__323,axiom,
    ! [Uu: real] : aa_real_real(aTP_Lamm_cr,Uu) = zero_zero_real ).

% ATP.lambda_323
tff(fact_10529_ATP_Olambda__324,axiom,
    ! [Uu: rat] : aa_rat_rat(aTP_Lamm_cs,Uu) = zero_zero_rat ).

% ATP.lambda_324
tff(fact_10530_ATP_Olambda__325,axiom,
    ! [Uu: nat] : aa_nat_complex(aTP_Lamm_hj,Uu) = zero_zero_complex ).

% ATP.lambda_325
tff(fact_10531_ATP_Olambda__326,axiom,
    ! [Uu: nat] : aa_nat_real(aTP_Lamm_hk,Uu) = zero_zero_real ).

% ATP.lambda_326
tff(fact_10532_ATP_Olambda__327,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_ct,Uu) = zero_zero_nat ).

% ATP.lambda_327
tff(fact_10533_ATP_Olambda__328,axiom,
    ! [Uu: nat] : aa_nat_int(aTP_Lamm_hl,Uu) = zero_zero_int ).

% ATP.lambda_328
tff(fact_10534_ATP_Olambda__329,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_cu,Uu) = zero_zero_int ).

% ATP.lambda_329
tff(fact_10535_ATP_Olambda__330,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_cg,Uu)
    <=> $false ) ).

% ATP.lambda_330
tff(fact_10536_ATP_Olambda__331,axiom,
    ! [Uu: complex] :
      ( aa_complex_o(aTP_Lamm_cf,Uu)
    <=> $false ) ).

% ATP.lambda_331
tff(fact_10537_ATP_Olambda__332,axiom,
    ! [Uu: real] :
      ( aa_real_o(aTP_Lamm_ch,Uu)
    <=> $false ) ).

% ATP.lambda_332
tff(fact_10538_ATP_Olambda__333,axiom,
    ! [Uu: $o] :
      ( aa_o_o(aTP_Lamm_ci,(Uu))
    <=> $false ) ).

% ATP.lambda_333
tff(fact_10539_ATP_Olambda__334,axiom,
    ! [Uu: nat] :
      ( aa_nat_o(aTP_Lamm_cj,Uu)
    <=> $false ) ).

% ATP.lambda_334
tff(fact_10540_ATP_Olambda__335,axiom,
    ! [Uu: int] :
      ( aa_int_o(aTP_Lamm_ck,Uu)
    <=> $false ) ).

% ATP.lambda_335
tff(fact_10541_ATP_Olambda__336,axiom,
    ! [Uu: vEBT_VEBTi] : aa_VEB205140289345309584at_nat(aTP_Lamm_dx,Uu) = suc ).

% ATP.lambda_336
tff(fact_10542_ATP_Olambda__337,axiom,
    ! [Uu: vEBT_VEBT] : aa_VEB1122092716774696213at_nat(aTP_Lamm_dt,Uu) = suc ).

% ATP.lambda_337
tff(fact_10543_ATP_Olambda__338,axiom,
    ! [Uu: real] : aa_real_fun_nat_nat(aTP_Lamm_du,Uu) = suc ).

% ATP.lambda_338
tff(fact_10544_ATP_Olambda__339,axiom,
    ! [Uu: $o] : aa_o_fun_nat_nat(aTP_Lamm_dv,(Uu)) = suc ).

% ATP.lambda_339
tff(fact_10545_ATP_Olambda__340,axiom,
    ! [Uu: nat] : aa_nat_fun_nat_nat(aTP_Lamm_dw,Uu) = suc ).

% ATP.lambda_340

% Helper facts (15)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X7: $o,Y6: $o] :
      ( ( (X7) != (Y6) )
      | aa_o_o(fequal_o((X7)),(Y6)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X7: $o,Y6: $o] :
      ( ~ aa_o_o(fequal_o((X7)),(Y6))
      | ( (X7) = (Y6) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X7: int,Y6: int] :
      ( ( X7 != Y6 )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X7),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X7: int,Y6: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X7),Y6)
      | ( X7 = Y6 ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X7: nat,Y6: nat] :
      ( ( X7 != Y6 )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X7),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X7: nat,Y6: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X7),Y6)
      | ( X7 = Y6 ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X7: real,Y6: real] :
      ( ( X7 != Y6 )
      | aa_real_o(aa_real_fun_real_o(fequal_real,X7),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X7: real,Y6: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(fequal_real,X7),Y6)
      | ( X7 = Y6 ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X8: real] : aa_real_o(P,X8) ) ) ).

tff(help_fequal_2_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X7: complex,Y6: complex] :
      ( ( X7 != Y6 )
      | aa_complex_o(fequal_complex(X7),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__Complex__Ocomplex_T,axiom,
    ! [X7: complex,Y6: complex] :
      ( ~ aa_complex_o(fequal_complex(X7),Y6)
      | ( X7 = Y6 ) ) ).

tff(help_fequal_2_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X7: vEBT_VEBT,Y6: vEBT_VEBT] :
      ( ( X7 != Y6 )
      | aa_VEBT_VEBT_o(fequal_VEBT_VEBT(X7),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X7: vEBT_VEBT,Y6: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(fequal_VEBT_VEBT(X7),Y6)
      | ( X7 = Y6 ) ) ).

tff(help_fequal_2_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X7: product_prod_int_int,Y6: product_prod_int_int] :
      ( ( X7 != Y6 )
      | aa_Pro4592068669816069595_int_o(fequal6010348034681699726nt_int(X7),Y6) ) ).

tff(help_fequal_1_1_fequal_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X7: product_prod_int_int,Y6: product_prod_int_int] :
      ( ~ aa_Pro4592068669816069595_int_o(fequal6010348034681699726nt_int(X7),Y6)
      | ( X7 = Y6 ) ) ).

% Conjectures (11)
tff(conj_0,hypothesis,
    tia = aa_VEB5586453056816421800_VEBTi(vEBT_Nodei(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))))),aa_nat_nat(suc,aa_nat_nat(suc,va)),x13),x14) ).

tff(conj_1,hypothesis,
    x11 = some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))))) ).

tff(conj_2,hypothesis,
    ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),mi) ).

tff(conj_3,hypothesis,
    size_s7982070591426661849_VEBTi(tree_is) = size_s6755466524823107622T_VEBT(treeList) ).

tff(conj_4,hypothesis,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))) != mi ).

tff(conj_5,hypothesis,
    xa = mi ).

tff(conj_6,hypothesis,
    vEBT_vebt_mint(aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) = some_nat(y) ).

tff(conj_7,hypothesis,
    ma = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary)))))) ).

tff(conj_8,hypothesis,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),size_s6755466524823107622T_VEBT(treeList)) ).

tff(conj_9,hypothesis,
    ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(summary))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

tff(conj_10,conjecture,
    entails(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,aa_nat_VEBT_VEBT(nth_VEBT_VEBT(list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),nth_VEBT_VEBTi(list_u6098035379799741383_VEBTi(tree_is,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),xb),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),aa_lis2508312872377092907i_assn(snga_assn_VEBT_VEBTi(x13),list_u6098035379799741383_VEBTi(tree_is,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),xb)))),aa_VEBT_VEBTi_assn(aa_VEB3717855479479287634i_assn(vEBT_vebt_assn_raw,summary),x14))),aa_lis2508312872377092907i_assn(vEBT_L1528199826722428489_VEBTi(minus_minus_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),size_s6755466524823107622T_VEBT(treeList)),insert_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),bot_bot_set_nat)),vEBT_vebt_assn_raw,list_u1324408373059187874T_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_VEBT_VEBT(vEBT_vebt_delete(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),list_u6098035379799741383_VEBTi(tree_is,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),the_nat(vEBT_vebt_mint(summary)))),the_nat(vEBT_vebt_mint(aa_nat_VEBT_VEBT(nth_VEBT_VEBT(treeList),the_nat(vEBT_vebt_mint(summary))))))),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,va),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),xb))),aa_assn_assn(times_times_assn(aa_assn_assn(times_times_assn(aa_d_11_ATP_assn(aa_c_13644421539718531832P_assn(a_11_ATP,nth_c_11_ATP(uu_16_ATP,uua_16_ATP)),xi_11_ATP)),vEBT_L375988980963497884911_ATP(minus_minus_set_nat(i_11_ATP,insert_nat(i_11_ATP2,bot_bot_set_nat)),a_11_ATP,xs_11_ATP,xsi_11_ATP))),f_11_ATP)) ).

%------------------------------------------------------------------------------